Kangro-100kangro100.ut.ee/kangro100-abstract_book.pdf · Contents Sponsors and partners 7 Scienti c and organizing committee 9 Plenary speakers 10 Programs 11 Program for participants

Kangro-100Methods of Analysis and Algebra

International conference dedicated to the centennial ofProfessor Gunnar Kangro

Tartu, EstoniaSeptember 1–6, 2013

Bookof Abstracts

Estonian Mathematical SocietyTartu2013

Kangro-100Methods of Analysis and Algebra

International conference dedicated to theCentennial of Professor Gunnar Kangro

Tartu, EstoniaSeptember 1–6, 2013

Book of AbstractsEdited by

E. Ojaand

K. Kaarli, A. Lissitsin, K. Mikkor, M. Põldvere, I. Zolk

ISBN 978-9949-9180-6-5

Copyright by Institute of Mathematics, University of Tartu,and Estonian Mathematical Society, 2013

Editors:

Eve OjaFaculty of Mathematics and Computer Science, University of Tartu, 50409 Tartu, J. Liivi 2,Estonia;Estonian Academy of Sciences, 10130 Tallinn, Kohtu 6, [email protected]

Kalle Kaarli, Aleksei Lissitsin, Kristel Mikkor, Märt Põldvere, Indrek ZolkFaculty of Mathematics and Computer Science, University of Tartu, 50409 Tartu, J. Liivi 2,[email protected]@[email protected]@[email protected]

Contents

Sponsors and partners 7

Scientific and organizing committee 9

Plenary speakers 10

Programs 11Program for participants. Time schedule . . . . . . . . . . . . . . . . . . . . . . 11Program for accompanying persons. Time schedule . . . . . . . . . . . . . . . . 21Social events and excursions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

Participants 27

Abstracts 39K. Kaarli, T. Leiger, and A. Tali, Life and work of Professor Gunnar Kangro

(21.11.1913–25.12.1975) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39A. Aasma, Characterization of A-statistical convergence with speed . . . . . 42T. A. Abrahamsen, J. Langemets, and V. Lima, Uniformly square Banach

spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43K. Ain, R. Lillemets, and E. Oja, Compactness defined by ℓp-spaces . . . . . 44S. Akduman, I-Ward continuity . . . . . . . . . . . . . . . . . . . . . . . . . 45R. M. Aron, The Bishop–Phelps–Bollobás theorem for operators, and Lin-

denstrauss properties A and B . . . . . . . . . . . . . . . . . . . . . . . . . 46H. Bendová, Quantitative Grothendieck property . . . . . . . . . . . . . . . 47J. Boos, Hahn spaces in Fréchet spaces . . . . . . . . . . . . . . . . . . . . . 48R. del Campo, A. Fernández, A. Manzano, F. Mayoral, and F. Naranjo,

Hardy inequalities and interpolation of Lorentz spaces associated to a vec-tor measure (I) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

İ. Çanak and Ü. Totur, An extended Tauberian theorem for the weightedmean method of summability . . . . . . . . . . . . . . . . . . . . . . . . . 51

B. Cascales, Radon–Nikodým indexes and measures of weak noncompactness 53T. Ciaś, Commutative subalgebras of the algebra of smooth operators . . . . 54J. Cırulis, On the Hermitian part of Rickart *-rings . . . . . . . . . . . . . . 56H. G. Dales, Multi-norms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

3

J. M. Delgado and C. Piñeiro, A characterization of operators with p-summingadjoint via p-limited sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

J. Diestel, Sums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59D. Dudzik, An outer measure on a commutative ring . . . . . . . . . . . . . . 60A. Fernández, F. Mayoral, and F. Naranjo, Different kinds of integrals in

the same formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61G. Godefroy, Approximation properties in Banach spaces . . . . . . . . . . . 62I. Bloshanskii and D. Grafov, “Almost” Cauchy property for the sequence of

partial sums of Fourier series of functions in Lp, p > 1 . . . . . . . . . . . 63J. Grygiel and K. Grygiel, On blocks of skeleton tolerances . . . . . . . . . . 65P. Hájek and T. Schlumprecht, The Szlenk index of Lp(X) . . . . . . . . . . 66J. Henno, Structure of generating sets for reversible computations . . . . . . 67D. Israfılov, Approximation in the weighted Lebesgue spaces . . . . . . . . . 70G. Janelidze, Nice categories . . . . . . . . . . . . . . . . . . . . . . . . . . . 72E. Jiménez Fernández, M. A. Juan, and Enrique A. Sánchez-Pérez, Regular

method of summability and the weak Fatou property . . . . . . . . . . . . 73M. Johanson, Unconditional ideals of compact operators . . . . . . . . . . . 74W. Johnson, Approximation properties of a Banach space and its subspaces 75K. Kaarli, Compatible function extension and majority functions . . . . . . 76K. A. Kearnes, Finitely based finite algebras . . . . . . . . . . . . . . . . . . 77A. Kivinukk, On summability of orthogonal expansions and Shannon sam-

pling series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78O. Košik and P. Mayr, Term-equivalence of semilattices . . . . . . . . . . . . 79O. I. Kuznetsova and A. N. Podkorytov, On strong averages of spherical

Fourier sums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80V. Laan, On congruence extension property for ordered algebras . . . . . . . 82R. Haller, J. Langemets, and M. Põldvere, On duality of diameter 2 properties 83V. Laohakosol, The positivity problem for linear recurrence sequences . . . . 84S. Bloshanskaya, I. Bloshanskii, and O. Lifantseva, Multiple Fourier expan-

sions over Walsh–Paley and trigonometric systems . . . . . . . . . . . . . 85E. Liflyand, Fourier transform versus Hilbert transform . . . . . . . . . . . . 87R. Lillemets, Some properties of generating systems of sets and sequences . . 89M. Lindström, Essential norm estimates for composition operators on BMOA 90H. Machida, Remarks on essentially minimal clones . . . . . . . . . . . . . . 91E. Malkowsky, Measures of noncompactness and some applications . . . . . 92L. Márki, Commutative orders in semigroups . . . . . . . . . . . . . . . . . 93M. Mathieu, Derivations and local multipliers of C*-algebras . . . . . . . . . 94R. del Campo, A. Fernández, A. Manzano, F. Mayoral, and F. Naranjo,

Hardy inequalities and interpolation of Lorentz spaces associated to a vec-tor measure (II) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

P. Mayr, The degree of operations on groups . . . . . . . . . . . . . . . . . . 97T. Metsmägi, The variation detracting property of some Shannon sampling

series and their derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

4

V. Montesinos, A note on extreme points of C∞-smooth balls in polyhedralspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

R. del Campo, A. Fernández, F. Mayoral, and F. Naranjo, Complex inter-polation of Lp-spaces of integrable functions with respect to vector measures100

H. Niglas, E. Oja, and I. Zolk, Lipschitz functions and M -ideals . . . . . . . 101A. Novikova, Lyapunov theorem for q-concave Banach spaces . . . . . . . . . 102O. Nygaard, The symmetric strong diameter 2 property in Banach spaces . . 103O. Orlova and G. Tamberg, On approximation properties of Kantorovich-

type sampling operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104K. Pavlyk, Productive property in topological inverse semigroups . . . . . . . 106D. Pellegrino, On the Bohnenblust–Hille inequality . . . . . . . . . . . . . . 107M. Abel and R. M. Pérez-Tiscareño, Bornological algebras which induce a

topology that gives a topological algebra structure . . . . . . . . . . . . . 108A. Pietsch, Traces of operators and their history . . . . . . . . . . . . . . . . 109U. Pintoptang, Necklaces and q-cycles . . . . . . . . . . . . . . . . . . . . . 110A. Plichko, Two approximation properties . . . . . . . . . . . . . . . . . . . . 111A. Przestacki, Closed range composition operators for one-dimensional smooth

symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112H.-J. Rack, Extensions of Schur’s inequality for the leading coefficient of

bounded polynomials with one prescribed zero . . . . . . . . . . . . . . . . 113O. I. Reinov, Eigenvalues of (r, p)-nuclear operators and approximation prop-

erties of order (r, p) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115K. Rozhinskaya and I. Zolk, On M(a,B, c)-ideals in Banach spaces . . . . . 116P. Rueda, Tensor products of Lp-spaces and applications to dominated poly-

nomials and spaces of polynomials . . . . . . . . . . . . . . . . . . . . . . 117A. Saksa and A. Kivinukk, Approximation by trigonometric Blackman- and

Rogosinski-type operators . . . . . . . . . . . . . . . . . . . . . . . . . . . 118E. A. Sánchez-Pérez, Factorization of homogeneous maps between Banach

function spaces and applications . . . . . . . . . . . . . . . . . . . . . . . . 119E. Savas, Iλ-statistically convergent sequences in topological groups . . . . . 120A. Šeletski and A. Tali, On strong and statistical convergences

in some families of summability methods . . . . . . . . . . . . . . . . . . . 121S. A. Sezer and İ. Çanak, Tauberian remainder theorems for the (N, p)

summability method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123M. Skopina, Around sampling theorem . . . . . . . . . . . . . . . . . . . . . 124D. Skorokhodov, On the best approximation of functions from Hölder classes

by a subset of linear finite-rank positive methods . . . . . . . . . . . . . . 125M. Skrzyński, Remarks on rank functions and rank varieties . . . . . . . . . 127N. Sohail, Epimorphisms in certain categories of partially ordered semigroups 129A. Sołtysiak, Ideals with at most countable hull in certain algebras of analytic

functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130A. Šostak, Joint continuity versus separate continuity: on a class of Namioka

spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

5

Á. Szendrei, Growth rates of finite algebras . . . . . . . . . . . . . . . . . . . 132G. Tamberg, Approximation error of generalized Shannon sampling operators

with bandlimited kernels in terms of an averaged modulus of smoothness . 133V. Laohakosol and P. Tangsupphathawat, Algebraic cosine values at ratio-

nal multiples of π . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135E. Oja and S. Treialt, Some duality results on bounded approximation prop-

erties of pairs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137S. Lassalle and P. Turco, The KA- and the KA-uniform approximation property138H.-O. Tylli, Survey of vector-valued analytic composition operators . . . . . 139G. Vainikko, Error estimates for cardinal spline interpolation and quasi-

interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140S. Veldsman, Restricted reversible rings . . . . . . . . . . . . . . . . . . . . . 141M. Volkov, Matrix identities involving multiplication and transposition . . . 142M. Zeltser, On the Hahn property of bounded domains of special matrix

methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143V. Zhuk, Inequalities of the generalized Jackson theorem type for best ap-

proximations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144E. Oja and I. Zolk, The asymptotically commuting bounded approximation

property of Banach spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

Index 147

Notes 149

6

Sponsors and partners

• Enterprise Estonia

• Estonian Doctoral School in Mathematics and Statistics

• University of Tartu

• University of Tartu, Faculty of Mathematics and Computer Science

FACULTY OF MATHEMATICS AND COMPUTER SCIENCE

• University of Tartu, Institute of Mathematics

7

• Estonian Academy of Sciences

• Estonian Mathematical Society

• Estonian Research Council

• Estonian Ministry of Education and Research

• The Council of Gambling Tax

• Tartu City Government

• Tartu County Tourism Foundation

8

Scientific and organizing committee

• Eve Oja (chair), University of Tartu, Estonia

• Kalle Kaarli (co-chair), University of Tartu, Estonia

• Kristel Mikkor (co-chair), University of Tartu, Estonia

• Richard M. Aron, Kent State University, USA

• Janis Cırulis, University of Latvia, Latvia

• Rainis Haller, University of Tartu, Estonia

• Andi Kivinukk, Tallinn University, Estonia

• Ulrich Knauer, Carl von Ossietzky Universität, Germany

• Valdis Laan, University of Tartu, Estonia

• Aleksei Lissitsin, University of Tartu, Estonia

• Olav Nygaard, University of Agder, Norway

• Märt Põldvere, University of Tartu, Estonia

• Anne Tali, Tallinn University, Estonia

• Hans-Olav Tylli, University of Helsinki, Finland

• Dirk Werner, Freie Universität Berlin, Germany

• Indrek Zolk, University of Tartu, Estonia

• Svetlana Saprõkova (secretary), University of Tartu, Estonia

9

Plenary speakers

• Richard M. Aron, Kent State University, USA

• Johann Boos, University of Hagen, Germany

• Bernardo Cascales, University of Murcia, Spain

• Harold Garth Dales, University of Lancaster, UK

• Joe Diestel, Kent State University, USA

• Gilles Godefroy, Institute of Mathematics of Jussieu, France

• George Janelidze, University of Cape Town, South Africa

• William Johnson, Texas A&M University, USA

• Keith Kearnes, University of Colorado, USA

• Ralph McKenzie, Vanderbilt University, USA

• Albrecht Pietsch, Friedrich Schiller University, Germany

• Ágnes Szendrei, University of Colorado, USA

• Mikhail Volkov, Ural Federal University, Russia

10

Programs

Program for participants. Time schedule

SUNDAY, SEPTEMBER 1, 2013

Tallinn11.00–13.00 Walking excursion in Tallinn (see p. 24)

Science Centre AHHAASadama 1, TartuHall:15.00–18.00 Registration

History Museum of the University of TartuLossi 25, Toome Hill, Tartu18.00–21.00 Welcome Reception (see p. 24)

11

MONDAY, SEPTEMBER 2, 2013

Science Centre AHHAASadama 1, Tartu

Hall:8.00–9.00 Registration

Lecture room Lektoorium:9.00–9.15 Opening

Welcome to Estonia

Chair: Albrecht Pietsch9.20–9.50 Anne Tali

Life and Work of Professor Gunnar Kangro10.00–10.50 Richard M. Aron

The Bishop–Phelps–Bollobás theorem for operators, and Lindenstraussproperties A and B

Coffee break

Chair: Keith Kearnes11.10–12.00 Ralph McKenzie

Weird equations, tame congruences and crazy, wonderful people

Lunch break

Lecture room Lektoorium: Lecture room Laika: Lecture room Apollo:

Analysis I Algebra/Analysis I Analysis II

Chair: Bernardo Cascales Chair: Mikhail Volkov Chair: Johann Boos

13.00–13.30 Petr HájekThe Szlenk index ofLp(X)

László MárkiCommutative orders insemigroups

Maria SkopinaAround sampling theorem

13.35–14.05 Martin MathieuDerivations and localmultipliers of C*-algebras

Peter MayrThe degree of operationson groups

Eberhard MalkowskyMeasures of noncompact-ness and some applica-tions

14.10–14.35 Daniel PellegrinoOn the Bohnenblust–Hilleinequality

Katarzyna GrygielOn blocks of skeletontolerances

Eduardo Jiménez Fer-nándezRegular method ofsummability and theweak Fatou property

Coffee break

12

Chair: Anatolij Plichko Chair: Hans-Olav Tylli Chair: Ekrem Savas

15.10–15.35 Andrzej SołtysiakIdeals with at mostcountable hull in cer-tain algebras of analyticfunctions

Marcin SkrzyńskiRemarks on rank func-tions and rank varieties

Olga I. KuznetsovaOn strong averages ofspherical Fourier sums

15.40–16.05 Enrique A. Sánchez-PérezFactorization of homo-geneous maps betweenBanach function spacesand applications

Reyna María Pérez-TiscareñoBornological algebraswhich induce a topologythat gives a topologicalalgebra structure

Ants AasmaCharacterization of A-statistical convergencewith speed

16.10–16.35 Kati AinCompactness defined byℓp-spaces

Tomasz CiaśCommutative subalgebrasof the algebra of smoothoperators

İbrahim ÇanakAn extended Tauberiantheorem for the weightedmean method of summa-bility

16.40–17.00 Juan Manuel DelgadoA characterization ofoperators with p-summingadjoint via p-limited sets

Adam PrzestackiClosed range compo-sition operators forone-dimensional smoothsymbols

Sefa Anıl SezerTauberian remaindertheorems for the (N, p)

summability method

18.00–20.00 Walking excursion in Tartu (see p. 24)

13

TUESDAY, SEPTEMBER 3, 2013


Lecture room Lektoorium:8.45–9.00 Welcome to Estonia

Chair: Ralph McKenzie9.00–9.50 Albrecht Pietsch

Traces of operators and their history10.00–10.50 Mikhail Volkov

Matrix identities involving multiplication and transpositionCoffee break

Chair: Anne Tali11.10–12.00 Johann Boos

Hahn spaces in Fréchet spaces

Lunch break

13.00–13.45 Poster Session in the Hall:

Marje JohansonUnconditional ideals of compact operators

Oleg KošikTerm-equivalence of semilattices

Johann LangemetsOn duality of diameter 2 properties

Rauni LillemetsSome properties of generating systems of sets and sequences

Heiki NiglasLipschitz functions and M -ideals

Anna NovikovaLyapunov theorem for q-concave Banach spaces

Olga OrlovaOn approximation properties of Kantorovich-type sampling operators

Ksenia RozhinskayaOn M(a,B, c)-ideals in Banach spaces

Anna SaksaApproximation by trigonometric Blackman- and Rogosinski-type operators

Silja TreialtSome duality results on bounded approximation properties of pairs

Indrek ZolkThe asymptotically commuting bounded approximation property of Banach spaces

14



Chair: Gilles Godefroy Chair: Ágnes Szendrei Chair: Andi Kivinukk

13.50–14.20 Hans-Olav TylliSurvey of vector-valuedanalytic compositionoperators

Hajime MachidaRemarks on essentiallyminimal clones

Elijah LiflyandFourier transform versusHilbert transform

14.25–14.55 Vicente MontesinosA note on extreme pointsof C∞-smooth balls inpolyhedral spaces

Stefan VeldsmanRestricted reversible rings

Gennadi VainikkoError estimates for car-dinal spline interpolationand quasiinterpolation

15.00–15.30 Anatolij PlichkoTwo approximationproperties

Janis CırulisOn the Hermitian part ofRickart *-rings

Ekrem SavasIλ-statistically convergentsequences in topologicalgroups

Coffee break

Chair: Petr Hájek Chair: Ágnes Szendrei Chair: Maria Skopina

16.05–16.30 Mikael LindströmEssential norm estimatesfor composition operatorson BMOA

Kalle KaarliCompatible functionextension and majorityfunctions

Vladimir ZhukInequalities of the gener-alized Jackson theoremtype for best approxima-tions

Chair: Antonio Fernán-dez

16.35–17.00 Pilar RuedaTensor products of Lp-spaces and applicationsto dominated polynomialsand spaces of polynomials

Ricardo del CampoHardy inequalities andinterpolation of Lorentzspaces associated to avector measure (I)

Dmytro SkorokhodovOn the best approxima-tion of functions fromHölder classes by a sub-set of linear finite-rankpositive methods

17.05–17.30 Oleg I. ReinovEigenvalues of (r, p)-nuclear operators andapproximation propertiesof order (r, p)

Fernando MayoralHardy inequalities andinterpolation of Lorentzspaces associated to avector measure (II)

Gert TambergApproximation errorof generalized Shannonsampling operators withbandlimited kernels interms of an averagedmodulus of smoothness

15

17.35–17.55 Pablo TurcoThe KA- and the KA-uniform approximationproperty

Setenay AkdumanI-Ward continuity

Tarmo MetsmägiThe variation detractingproperty of some Shan-non sampling series andtheir derivatives

Lecture room Lektoorium:18.00–19.00 The ethics of publishing and open access

Harold Garth Dales; open discussionHarold Garth Dales is Vice-Chairman of the Ethics Committee of the European Mathe-matical Society. This committee is mainly concerned with ethical issues concerned withthe publishing of mathematics; a Code of Practice can be found on the website of theEMS. Garth will introduce the Code briefly and invite comments. Input from editorsand committee members of national societies is particularly welcome.A new issue seems likely to be relevant in the coming years: several governments and theEU are promoting “open access” to scientific journals. This gives rise to further problemsand some ethical issues, and the Committee will have to come to grip with these. Wewelcome discussion and input of information from those who know what is happening intheir own (or other) countries.

16

WEDNESDAY, SEPTEMBER 4, 2013

Science Centre AHHAASadama 1, TartuLecture room Lektoorium:8.45–9.00 Welcome to EstoniaChair: William Johnson9.00–9.50 Joe Diestel

Sums10.00–10.50 Ágnes Szendrei

Growth Rates of Finite Algebras

13.00–21.00 Excursions to the Lake Peipsi Region, the Haanja Nature Park in VõruCounty, the Soomaa National Park and the town Viljandi (see p. 24)

17

THURSDAY, SEPTEMBER 5, 2013


Lecture room Lektoorium:8.45–9.00 Welcome to Estonia

Chair: George Janelidze9.00–9.50 Harold Garth Dales

Multi-norms10.00–10.50 Keith Kearnes

Finitely based finite algebrasCoffee break

Chair: Joe Diestel11.10–12.00 Bernardo Cascales

Radon–Nikodým indexes and measures of weak noncompactness

Lunch break



Chair: Vicente Mon-tesinos

Chair: Janis Cırulis Chair: Elijah Liflyand

13.00–13.25 Antonio FernándezDifferent kinds of inte-grals in the same formula

Pinthira Tangsup-phathawatAlgebraic cosine values atrational multiples of π

Maria ZeltserOn the Hahn propertyof bounded domains ofspecial matrix methods

13.30–13.55 Hana BendováQuantitativeGrothendieck property

Kateryna PavlykProductive propertyin topological inversesemigroups

Heinz-Joachim RackExtensions of Schur’sinequality for the leadingcoefficient of boundedpolynomials with oneprescribed zero

14.00–14.25 Olav NygaardThe symmetric strongdiameter 2 property inBanach spaces

Nasir SohailEpimorphisms in certaincategories of partiallyordered semigroups

Anna ŠeletskiOn strong and statisticalconvergences in somefamilies of summabilitymethods

14.30–14.55 Trond A. AbrahamsenUniformly square Banachspaces

Valdis LaanOn congruence exten-sion property for orderedalgebras

Dariusz DudzikAn outer measure on acommutative ring

Coffee break

18

Chair: Martin Mathieu Chair: László Márki Chair: EberhardMalkowsky

15.30–15.55 Francisco NaranjoComplex interpolation ofLp-spaces of integrablefunctions with respect tovector measures

Vichian LaohakosolThe positivity problemfor linear recurrencesequences

Andi KivinukkOn summability of or-thogonal expansions andShannon sampling series

16.00–16.25 Daniyal IsrafılovApproximation in theweighted Lebesgue spaces

Umarin PintoptangNecklaces and q-cycles

Olga LifantsevaMultiple Fourier expan-sions over Walsh–Paleyand trigonometric sys-tems

16.30–16.55 Alexander ŠostakJoint continuity versusseparate continuity: on aclass of Namioka spaces

Jaak HennoStructure of generat-ing sets for reversiblecomputations

Denis Grafov“Almost” Cauchy propertyfor the sequence of partialsums of Fourier series offunctions in Lp, p > 1

Science Centre AHHAASadama 1, Tartu19.00–23.00 Conference Banquet (see p. 26)

19

FRIDAY, SEPTEMBER 5, 2013

Science Centre AHHAASadama 1, TartuLecture room Lektoorium:8.45–9.00 Welcome to EstoniaChair: Richard M. Aron9.00–9.50 William Johnson

Approximation properties of a Banach space and its subspaces10.00–10.50 George Janelidze

Nice categoriesCoffee breakChair: Harold Garth Dales11.10–12.00 Gilles Godefroy

Approximation properties in Banach spaces12.00–12.15 Closing


20

Program for accompanying persons. Time schedule




History Museum of the University of TartuLossi 25, Toome Hill, Tartu18.00–21.00 Welcome Reception (see p. 24)



Lecture room Lektoorium:9.00–9.15 Opening

Welcome to Estonia

9.20–9.50 Anne TaliLife and Work of Professor Gunnar Kangro

Estonian National MuseumJ. Kuperjanovi 9, Tartu11.00–13.00 Program “Who is an Estonian?”

Guided tour in the Estonian National Museum to learn about Estoniancustoms, clothes, and activities.

NB! Walk to the Estonian National Museum starts in front of the Science Centre AH-HAA at 10.30.15.00–16.30 Bus excursions in Tartu

Bus tour through districts of Tartu located further away from the oldtown and each having different type of planning, architecture, and his-tory.

NB! The bus departs in front of the Science Centre AHHAA.

18.00–20.00 Walking excursion in Tartu (see p. 24)

21

TUESDAY, SEPTEMBER 3, 2013

Science Centre AHHAASadama 1, TartuLecture room Lektoorium:

8.45–9.00 Welcome to Estonia

Estonian Agricultural MuseumPargi 4, Ülenurme, Tartu County10.00–13.00 Hands-on program “Rye bread on our table”

Workshop to learn about the history of agriculture in Estonia, process ofmaking bread and butter, and importance of rye bread as a traditionalEstonian food. Opportunity to make rye bread and butter yourself andtaste their delicious combination.

NB! Bus to the museum departs in front of the Science Centre AHHAA at 9.30.

Botanical Garden of the University of TartuLai 38, Tartu15.00–16.30 Walk with guide

Interesting tour in the university’s Botanical Garden to get to knowdifferent plants and flowers.

NB! Excursion starts from the gates of the Botanical Garden.


Science Centre AHHAASadama 1, TartuLecture room Lektoorium:

8.45–9.00 Welcome to Estonia

13.00–21.00 Excursions to the Lake Peipsi Region, the Haanja Nature Park in VõruCounty, the Soomaa National Park and the town Viljandi (see p. 24)


Science Centre AHHAASadama 1, TartuLecture room Lektoorium:8.45–9.00 Welcome to Estonia10.00-14.00 Guided tour in the Science Centre AHHAA

AHHAA leads you to the science, exciting discoveries, adventure, usefulskills, and fun entertainment.

NB! Excursion starts in the Hall near the Kangro-100 registration table.

16.00–17.00 Guided tour on the Emajõgi RiverBoat tour on the Emajõgi River to see the town from water.

22

NB! Boat “Hilara” departs in front of the Restaurant “Atlantis”. For boarding, be thereno later than 15.50!

Science Centre AHHAASadama 1, Tartu19.00–23.00 Conference Banquet (see p. 26)


Science Centre AHHAASadama 1, TartuLecture room Lektoorium:8.45–9.00 Welcome to Estonia

St. Anthony’s CourtyardLutsu 5, Tartu9.30–11.30 Excursion in St. Anthony’s Courtyard and workshop in the Leather

HouseGuided tour in all the buildings of St. Anthony’s Courtyard – a placewhere art is created and loved and where genuine master craftsmen work.Opportunity for everyone to make a perfect gift – leather medallion orkeychain – by one’s own hands.

NB! Excursion starts from the entrance to the courtyard.

Science Centre AHHAASadama 1, TartuLecture room Lektoorium:12.00–12.15 Closing


23

Social events and excursions


11.00–13.00 Walking excursion in TallinnNB! Excursion to the Old Town of Tallinn starts from the Viru Gate, Viru street, Tallinn.

History Museum of the University of TartuLossi 25, Toome Hill, Tartu18.00–21.00 Welcome ReceptionThe Welcome Reception takes place in the White Hall of the History Museum of theUniversity of Tartu, located in the former University Library and Dome Cathedral. TheDome Cathedral of Tartu is one of the earliest and most spectacular specimens of Gothicbrick architecture in the Baltic countries. Construction of the cathedral started in the13th century and lasted with intervals until the beginning of the 16th century. At theend of the 15th century, the construction of two high towers on the western face of thechurch was completed. The cathedral was abandoned and destroyed by a war in theend of the 16th century. The plans of rebuilding the cathedral were never carried out.When the University of Tartu was reopened in 1802, the choir of the church was rebuiltto accommodate the University Library. After the library moved to its new location in1982, the building was handed over to the History Museum.


18.00–20.00 Walking excursion in TartuNB! Excursion to the Old Town of Tartu starts from the fountain “The Kissing Students”located at the Tartu Town Hall Square.


13.00–21.00 Excursions to the Lake Peipsi Region, the Haanja Nature Park in VõruCounty, the Soomaa National Park and the town Viljandi

NB! Buses depart in front of the Science Centre AHHAA.

Lake Peipsi regionSightseeing: We drive to Lake Peipsi – one of the largest lakes in Europe. The lakehas very significantly affected the life of the people living on the shore. In this oneregion we will meet three different cultures – Russian Orthodox Old Believers, traces ofthe German barons era, and Estonian cultural history. Russian Old Believers who in the17th century became separated from the official Russian Orthodox Church and emigratedto Estonia, live in their unique one-street village trying to preserve their religion, historyand traditions. A magnificent manor house – the Alatskivi Castle – offers a glimpse intothe life of German barons in the 1870s. The third side of the region shows the life ofEstonian peasants in the 19th century and introduces the literary heritage of Juhan andJakob Liiv.Walking distance: 2–3 very short walks (appr. 300 m)Clothes: casual (most visits are indoors), ladies need a scarf to cover their head while

24

visiting the church of Old Believers (if possible then it is recommended that nobody iswearing shorts either)Dinner: Estonian cuisine

Haanja Nature Park and the town VõruSightseeing: Estonia is mostly flat: over 60% of the country is lower than 50 meters andonly 10% is more than 100 m above the sea level. In the north east the only hills areman-made: cone-shaped solid waste dumps and plateaus, ash plateaus and black semi-coke heaps. South Estonia, especially around Võru, is the hilliest.The Haanja Nature Park is located in the Haanja Uplands, the highest area in Estonia(about 18% of the area lies higher than 250 m from sea level. The nature park (itsarea is about 17,000 ha) was established to secure the preservation, re-establishment anddevelopment of the landscapes and nature communities typical to the Haanja Uplandsand also the historically established way of life, i.e., farmhouses spread far apart, smallplots of land, pastures and grasslands and handicraft traditions due to poor arable land.Within the nature park there are the highest point of the Baltic States – Suur Munamägi(318 m) and the deepest lake in Estonia – Rõuge Suurjärv (38 m).The hilly landscape of South Estonia has also shaped the little town of Võru. Võru isknown for its wooden architecture, beautiful lake beach promenade, unique dialect andsports centres and is a beloved destination among nature and sports lovers.Walking distance: 3 walks (2 km, 500 m, 200 m) in nature and 1-hour walking tour intown (short distances)Clothes: casual, comfortable shoes and raincoat or umbrella (visits indoors and walkingoutdoors)Dinner: Estonian cuisine

Soomaa National Park and the town ViljandiSightseeing: The Estonian territory (45,215.4 sq km) is about the same size as Denmark’sor Holland’s, but its population is only 1.4 million. Therefore a lot of nature’s beautyis preserved here. Forests cover over one half of Estonia, and one third of the forestsare protected areas. Meadows in Estonia are rich in different species much more so thanin Western Europe, and when we are travelling around the countryside we can discoverthe variety and colour of the roadside meadows. Estonia is a country of a thousandlakes – there are about 1200 natural lakes whose water area exceeds 1 ha. About onefifth of Estonia is covered with marshes and bogs, and most of these are located in thecentral and eastern parts of the country. We can see all this variety of the nature whilevisiting the Soomaa National Park, founded in 1993 and being an extensive wildernessarea in southwestern Estonia, containing large peat-bogs and thick forests interwoven bynumerous rivers and the floodplains that border them.Viljandi is an ancient hilly town in southern Estonia topped by impressive ruins of a once-powerful Livonian Order castle. Founded in 1283, Viljandi was an important medievaltrade center and a member of the Hanseatic League. Cobbled streets, ancient oak treesand a beautiful lake conjure up a very special atmosphere. Viljandi is also the capital offolk music for most Estonians.Walking distance: 1 long walk (appr. 5 km) in nature and 1-hour walking tour in town

25

Clothes: casual, comfortable shoes for a longer walk and raincoat (mostly outdoors)Dinner: Estonian cuisine


Science Centre AHHAASadama 1, Tartu19.00–23.00 Conference BanquetThe Conference Banquet takes place in the new building of the Science Centre AHHAA,opened on May 7, 2011. The Science Centre AHHAA is the biggest science centre inthe Baltic States with almost three thousand square meters for exhibition introducingscience and technology in a fun and playful way. Giant energy machine, or Newton’sapple tree, the Hoberman sphere, the adventurous elevator journey inside the Earth,mini-labs, stock market, gift shop, planetarium and much more interesting can be foundhere. Participants of the Kangro-100 conference have the chance to see all the attractionsof AHHAA during the Conference Banquet. In fact, the first hour is for attractions only,and dinner is served starting at 20.00.


17.00–19.00 Walking excursion in TallinnNB! Excursion to the Old Town of Tallinn starts from the Viru Gate, Viru street, Tallinn.

NB! More information about Estonia can be found on www.visitestonia.com.

26

Participants

Ants [email protected]

Tallinn Univ. of Tech.Estonia

Mati [email protected] of Tartu

Estonia

Trond A. Abrahamsentrond.a.abrahamsen@

@uia.noUniversity of Agder

Norway

Kati [email protected]

University of TartuEstonia

Setenay Akdumansetenayakduman@

@gmail.comDokuz Eylul Univ.

Turkey

Richard MartinAron

[email protected] State University

USA

Hana Bendová[email protected] UniversityCzech Republic

Svetlana Bloshanskayaigorbloshanskii@

@rambler.ruNational Research

Nuclear Univ. MEPHIRussia

27

Johann Boosjohann.boos@

@fernuni-hagen.deUniversity of Hagen

Germany

Ricardo del [email protected]. of Seville

Spain

İbrahim Ç[email protected]

Ege UniversityTurkey

Bernardo [email protected]

University of MurciaSpain

Tomasz Ciaś[email protected]. Mickiewicz Univ.

in PoznańPoland

Janis Cı[email protected]

University of LatviaLatvia

Harold Garth [email protected]

Univ. of LancasterUnited Kingdom

Juan ManuelDelgado

[email protected]. of Seville

Spain

Joe [email protected] State University

USA

Dariusz Dudzikdariusz.dudzik@

@gmail.comPedagogical Univ.

of CracowPoland

Antonio Ferná[email protected]

Univ. of SevilleSpain

Gilles Godefroygodefroy@

@math.jussieu.frInst. of Math.

of JussieuFrance

28

Denis [email protected] StateRegional Univ.

Russia

Joanna [email protected] Dlugosz University

in CzestochowaPoland

Katarzyna Grygielgrygiel@

@tcs.uj.edu.plJagiell. Univ. in Cracow

Poland

Petr Há[email protected]

Czech TechnicalUniv. in PragueCzech Republic

Rainis [email protected] of Tartu

Estonia

Uno Hä[email protected] of Tartu

Estonia

Jaak [email protected]


Daniyal Israfilovmdaniyal@

@balikesir.edu.trBalikesir University

Turkey

George JanelidzeGeorge.Janelidze@

@uct.ac.zaUniv. of Cape Town

South Africa

EduardoJiménez Ferná[email protected]. de Granada

Spain

Marje Johansonmarje.johanson@

@ut.eeUniversity of Tartu

Estonia

William [email protected]

Texas A&M Univ.USA

29

Endel Jürimä[email protected] of Tartu

Estonia

Kalle [email protected]


Urve [email protected] of Tartu

Estonia

Keith KearnesKeith.Kearnes@@Colorado.edu

Univ. of ColoradoUSA

Mati [email protected] of Tartu

Estonia

Andi [email protected]

Tallinn UniversityEstonia

Enno [email protected] of Tartu

Estonia

Oleg Koš[email protected] of Tartu

Estonia

Olga Kuznetsovakuznets@

@iamm.ac.donetsk.uaInst. of Applied Math.and Mech. of NASU

Ukraine

Valdis [email protected]


Jüri [email protected]


Johann Langemetsjohann.langemets@


Estonia

30

Vichian [email protected]

Kasetsart Univ.Thailand

Qaisar Latiflubayeifr@

@googlemail.comAbdus Salam Schoolof Math. Sciences

Pakistan

Toivo [email protected] of Tartu

Estonia

Olga Lifantsevaov-lifantseva@

@yandex.ruMoscow StateRegional Univ.

Russia

Elijah [email protected] University

Israel

Rauni [email protected]


Mikael Lindströmmikael.lindstrom@

@oulu.fiUniversity of Oulu

Finland

Jüri Lippusjyri.lippus@@affecto.com

AffectoEstonia

Aleksei Lissitsinaleksei.lissitsin@


Estonia

Leiki [email protected] of Tartu

Estonia

Hajime Machidamachida@

@math.hit-u.ac.jpInternational

Christian Univ.Japan

EberhardMalkowsky

eberhard.malkowsky@@math.uni-giessen.de

Fatih UniversityTurkey

31

László Márkimarki.laszlo@@renyi.mta.hu

MTA Rényi InstituteHungary

Martin [email protected]

Queen’s Univ. BelfastUnited Kingdom

Fernando [email protected]. of Seville

Spain

Peter [email protected]

JKU LinzAustria

Ralph McKenzieralph.n.mckenzie@@vanderbilt.eduVanderbilt Univ.

USA

Martin [email protected]


Tarmo Metsmä[email protected] University

Estonia

Kristel [email protected] of Tartu

Estonia

Vicente Montesinosvicente.montesinos@

@gmail.comPolytechnic University

of ValenciaSpain

VolodymyrMykhaylyuk

[email protected] Nat. Univ.

Ukraine

Francisco [email protected]. of Seville

Spain

Niels JørgenNielsen

[email protected]. of South. Denmark

Denmark

32

Heiki [email protected]


Anna Novikovanovikova.anna18@

@gmail.comWeizmann Inst. of Sc.

Israel

Olav [email protected] of Agder

Norway

Eve [email protected]


Olga [email protected] Univ. of Tech.

Estonia

Kateryna [email protected]


Daniel [email protected]

Federal Universityof Paraíba

Brazil

Reyna MaríaPérez-Tiscareño

[email protected] of Tartu

Estonia

Albrecht [email protected]

FSU JenaGermany

Umarin [email protected]

Naresuan Univ.Thailand

Anatolij [email protected] Univ. of Tech.

Poland

Anatolii Podkorytova.podkorytov@@gmail.com

St. PetersburgState University

Russia

33

Märt Põ[email protected]


Adam [email protected]. Mickiewicz Univ.

in PoznańPoland

Peeter [email protected] of Tartu

Estonia

Rabia QureshiRabiaqureshi1981@

@yahoo.comNational Univ. of

Comp. and Emerging Sc.Pakistan

Heinz-Joachim Rackheinz-joachim.rack@

@drrack.comGermany

Ülo [email protected] of Tartu

Estonia

Elmar [email protected] of Tartu

Estonia

Oleg [email protected]

St. PetersburgState Univ.

Russia

Ksenia [email protected] of Tartu

Estonia

Pilar [email protected]. of Valencia

Spain

Anna [email protected] University

Estonia

Enrique A.Sánchez-Pérez

[email protected] University

of ValenciaSpain

34

Ekrem [email protected]

IstanbulCommerce Univ.

Turkey

Anna Š[email protected]


Sefa Anıl Sezersefaanilsezer@

@gmail.comEge University

Turkey

Maria Skopinaskopina@

@MS1167.spb.eduSt. PetersburgState Univ.

Russia

Dmytro Skorokhodovdmitriy.skorokhodov@

@gmail.comDnepropetrovskNational Univ.

Ukraine

Marcin Skrzyń[email protected] Univ. of Tech.

Poland

Lenka Slavíková[email protected] UniversityCzech Republic

Richard JamesSmith

[email protected]. Coll. Dublin

Ireland

Olexandr [email protected]

Chernivtsi Nat. Univ.Ukraine

Nasir [email protected]


Andrzej Soł[email protected]. Mickiewicz Univ.

in PoznańPoland

Virge Soomer01.08.1945––16.07.2013


35

Aleksandrs Š[email protected] of Latvia

Latvia

Ágnes SzendreiAgnes.Szendrei@@Colorado.edu

Univ. of ColoradoUSA

Anne [email protected]


Gert [email protected] Univ. of Tech.

Estonia

PinthiraTangsupphathawat

[email protected] Univ.

Thailand

Lauri [email protected]


Margus Tõ[email protected]


Laur [email protected]


Silja [email protected] of Tartu

Estonia

Pablo [email protected]

University ofBuenos Aires

Argentina

Heino Türnpuheino.tyrnpu@@gmail.com


Hans-Olav Tyllihojtylli@

@cc.helsinki.fiUniv. of Helsinki

Finland

36

Gennadi Vainikkogennadi.vainikko@


Estonia

Stefan [email protected] Qaboos Univ.

Oman

Mikhail VolkovMikhail.Volkov@

@usu.ruUral Federal Univ.

Russia

Maria [email protected]


Vladimir [email protected]

St. PetersburgState University

Russia

Indrek [email protected] of Tartu

Estonia

Andras Zsaka.zsak@

@dpmms.cam.ac.ukCambridge Univ.United Kingdom

37

38

Abstracts

LIFE AND WORKOF PROFESSOR GUNNAR KANGRO(21.11.1913–25.12.1975)

Kalle Kaarli, Toivo Leiger, and Anne Tali

“Professor Gunnar Kangro, member of the Estonian Academy of Sciences, was themost famous Estonian mathematician of his time. He was a world-class professionalin his main research area – summability theory. His excellent courses and textbooks inalgebra and analysis advocated the use of new theories developed in the first half of thetwentieth century, and led the transition of Estonian mathematics to modern basis. Hislecture courses in functional analysis gave also momentum to the research in numericalanalysis in Estonia. Having supervised 23 Cand. Sc. theses, he is fully considered thefounder of contemporary Estonian mathematical school.”

In our talk we will try to open this superlative summary of the life and the work ofprofessor Gunnar Kangro (given on the homepage of our conference) in a more detailedway, basing on the factual materials introduced below.

Gunnar Kangro was born in 1913 in Tartu, in the family of a successful buildingcontractor and architect. He graduated from the Tartu Real Gymnasium in 1931 andthe same year started his mathematics studies at the University of Tartu. His fatherFromhold Kangro had bankrupted by that time, and thus he was in serious economicdifficulties during his student years. After graduating from the university and passingthe regular military service he was employed by Tallinn University of Technology in 1936as an assistant. In 1938 he defended his master thesis on polynomial series written undersupervision of professor H. Jaakson. Soon after that he started working on his doctoralthesis, being supported by a 2-year scholarship 1940–1941. The scholarship was closeddown in the end of 1940, shortly after Estonia was incorporated into the Soviet Union.Kangro’s work on doctoral thesis was interrupted by the beginning of the war betweenthe Soviet Union and Germany. In July 1941 he was forced to join the Red Army. Thanksto the help of professor J. Nuut, he was transferred back to scientific work in February1942, first to Chelyabinsk, later to Moscow, where he continued his research on doctoral

39

thesis. In 1944, after the war was over in Estonia, he returned home and started hiswork at the University of Tartu as a mathematics docent (associate professor). In 1947he defended his doctoral thesis. In 1951 he was appointed to the position of mathematicsprofessor at the University of Tartu. He worked there as a professor and a head of chairtill the end of his life. First, in years 1952–1959 he was the head of the chair of geometry,and later the head of the chair of analysis. In 1961 he became a corresponding memberof the Estonian Academy of Sciences.

In 1951 he married Hilja Puskar who had just graduated from the faculty of medicine.Hilja and Gunnar got two daughters – Külliki (1954) and Anu (1957). Both of thembecame medical doctors following in their mother’s steps, and Cand. Sc. in medicinefollowing in their father’s steps.

Basic developments in mathematics during the first half of the last century involvedbig changes in teaching mathematics at universities. Gunnar Kangro had the mainresponsibility for these changes at the University of Tartu. Thus, he built up new moderncourses in algebra and analysis. He was one of the first professors in the Soviet Union whotaught the course of functional analysis which turned out to be a starting point for a newresearch direction in Estonia, the theory of numerical methods. Professor Gunnar Kangrobecame a recognized leader of the after-war-time mathematical community of Estonia.Under his leadership also the research directions, where he himself was not active (e.g.,numerical methods, algebra, geometry) were developed. He wrote the excellent universitytextbooks in algebra and analysis.

The research of G. Kangro himself was mainly carried out in the theory of summa-bility of series and sequences. In the early years of his work this theory was called thetheory of divergent series. His doctoral thesis was based on the ideas of É. Borel. Onthis basis he created the new convergence theory, which included the Borel’s theory andcould be applied in investigations of convergence of power series. He introduced newsummability methods and applied them for solving problems of convergence and analyticcontinuation of power series. In the nineteen fifties, in parallel with German mathemati-cians A. Peyerimhoff and W. Jurkat, G. Kangro built up a basis for systematic study ofsummability factors. It is characteristic to his research methodology that he combinedmethods of functional analysis with classical methods of analysis. Together with his stu-dent S. Baron he started describing summability factors of double series, thus giving animpulse for investigations of summability of double series.

The applications of summability theory to orthogonal series and problems relatedto Tauberian theorems motivated him to found in the years 1960–1970 the theory ofsummability with speed. Within the framework of this theory he succeeded in solvingseveral problems of the theory of summability and of the theory of functions. Basingthis new theory on functional analysis, he also pointed out the main possibilities of itsapplications. His work was interrupted by his death in 1975.Doubtless, the main contribution of professor Kangro was bringing up a new generationof Estonian mathematicians. Under his supervision 23 candidate (i.e., PhD) theses inmathematics were defended, majority of them in summability theory. This is why wecan talk about Tartu Summability School.

40

Professor G. Kangro was a great personality. Supervising his students in writing theirtheses he also formed their beliefs and ways of life. His students remember him with loveand deepest respect, carrying on his mathematical and human legacy.

Professor G. Kangro is buried in the Raadi Cemetery of Tartu.

Faculty of Mathematics and Computer Science, University of Tartu, 50409 Tartu, J. Liivi2, EstoniaE-mails: [email protected], [email protected] of Mathematics and Natural Sciences, Tallinn University, 10120 Tallinn, 25Narva mnt, EstoniaE-mail: [email protected]

41

Abstracts of Kangro-100, September 1–6, 2013, Tartu, Estonia

CHARACTERIZATIONOF A-STATISTICAL CONVERGENCE WITH SPEED

Ants Aasma

In [3], Kolk introduced the concept of A-statistical convergence for a non-negativeregular matrix A and studied matrix transforms of A-statistically convergent sequences.We introduce the notion of A-statistical convergence with speed. Defining the speed ofA-statistical convergence we use the notion of convergence of sequences with speed (wherethe speed was defined by a monotonically increasing positive sequence λ), introduced byKangro in [2]. Also we investigate the matrix transforms of A-statistically convergentsequences with speed. Let X and Y be sequence spaces. We study the matrix transformsfrom stλA ∩X → Y , where stλA is the set of all A-statistically convergent sequences withspeed λ. We show that, in the case when λ is bounded, our results cover some results ofKolk [3].

The research was supported by Estonian Science Foundation Grant 8627.

REFERENCES

[1] Ants Aasma, Matrix transforms of A-statistically convergent sequences with speed,FILOMAT (accepted).

[2] Gunnar Kangro, On the summability factors of the Bohr-Hardy type for a givenspeed I, Proc. Estonian Acad. Sci. Phys. Math. 18 (1969), 137–146.

[3] Enno Kolk, Matrix summability of statistically convergent sequences, Analysis 13(1993), 77–83.

Faculty of Economics, Tallinn University of Technology, 12618 Tallinn, 3 Akadeemiaroad, EstoniaE-mail: [email protected]

42


UNIFORMLY SQUARE BANACH SPACES

Trond A. Abrahamsen, Johann Langemets, and Vegard Lima

We say that a Banach space is locally uniformly square (LUS) if for every x in theunit sphere SX of X, there exists a sequence (yn) in the unit ball BX of X such that

‖x± yn‖ → 1 and ‖yn‖ → 1.

If X is LUS and the sequence (yn) tends to 0 weakly, we say that X is weakly uniformlysquare (ωUS).

We say that a Banach space is uniformly square (US) if for every x1, x2, . . . , xN ∈ SX ,there exists a sequence (yn) in BX such that

‖xi ± yn‖ → 1 for every i = 1, . . . , N, and ‖yn‖ → 1.

The motivation for studying such spaces is the fact that they possess properties whichin a sense are at the opposite side of the spectrum from the Radon–Nikodým property(any closed convex set has slices of arbitrarily small diameter). If X is

• LUS, then the diameter of every slice of BX is 2.

• ωUS, then the diameter of every non-empty relatively weakly open subset of BXis 2.

• US, then the diameter of every finite convex combination of slices of BX is 2.

Other basic properties of such spaces will be discussed as well.

REFERENCES

[1] D. Kubiak, Some geometric properties of the Cesàro function spaces, J. Conv. Anal.(to appear).

[2] J. J. Schäffer, Geometry of Spheres in Normed Spaces, Marcel Dekker Inc., NewYork, 1976, Lecture Notes in Pure and Applied Mathematics, No. 20. MR 0467256(57 #7120)

Department of Mathematics, University of Agder, Postbox 4224604 Kristiansand, Norway.E-mail: [email protected] of Mathematics, University of Tartu, J. Liivi 2, EE-50409 Tartu, Estonia.E-mail: [email protected]Ålesund University College, Postboks 1517, N-6025 Ålesund, Norway.E-mail: [email protected]

43


COMPACTNESS DEFINED BY ℓp-SPACES

Kati Ain, Rauni Lillemets, and Eve Oja

Let 1 ≤ p < ∞ and 1 ≤ r ≤ p∗, where p∗ is the conjugate index of p. We say that alinear operator T from a Banach space X to a Banach space Y is (p, r)-compact if theimage of the unit ball T (BX) is contained in ∑n anyn : (an) ∈ Bℓr (where (an) ∈ Bc0if r = ∞) for some p-summable sequence (yn) ∈ ℓp(Y ).

The p-compact operators, studied recently by J.M. Delgado, A.K. Karn, C. Piñeiro,E. Serrano, D.P. Sinha, and others, are precisely the (p, p∗)-compact operators.

We describe the quasi-Banach operator ideal structure of the class of (p, r)-compactoperators and how it relates to some other known operator ideals.

The research was partially supported by Estonian Science Foundation Grant 8976and Estonian Targeted Financing Project SF0180039s08.

REFERENCES

[1] K. Ain, R. Lillemets, and E. Oja, Compact operators which are defined by ℓp-spaces, Quaest. Math. 35 (2012), 145–159.

Faculty of Mathematics and Computer Science, University of Tartu, 50409 Tartu, J. Liivi2, EstoniaE-mails: [email protected], [email protected]

Faculty of Mathematics and Computer Science, University of Tartu, 50409 Tartu, J. Liivi2, Estonia; Estonian Academy of Sciences, Kohtu 6, 10130 Tallinn, EstoniaE-mail: [email protected]

44


I-WARD CONTINUITY

Setenay Akduman

A non-empty collection of subsets of a set X which is closed under the operations ofsubset and finite unions defines an ideal on X. The concept of I-convergence of sequencesof real numbers based on the notion of the ideal of subsets of N has been introducedin [1]. Then this concept has been extended to topological spaces and topological groupsas in the studies [2] and [6].

This study comes from the concept of Nθ-convergence which is introduced in [3] andthe concept of ideal convergence developed in some papers such as [4] and [5]. In [3], anew kind of continuity and a new kind of compactness have been introduced, namely,Nθ-ward continuity and Nθ-ward compactness. The concept of I-convergence encouragesus to approach these new types of convergence and compactness. This concept also hasbeen studied on topological groups in [6].

Our purpose is to generalize some studies on the new kinds of continuities and com-pactness to ideals. We will also examine the concept of I-convergence in topologicalgroups to extend some previous studies.

REFERENCES

[1] P. Kostyrko, M. Macaj, and T. Salat, Statistical convergence and I-convergence, Real Anal. Exch., submitted.

[2] B. K. Lahiri and P. Das, I and I∗-convergence in topological spaces, Math. Bo-hemica 130 (2005), 153–160.

[3] H. Cakalli, Nθ-ward continuity, In Abstract and Applied Anal., vol. 2012, HindawiPublishing Corporation.

[4] P. Kostyrko, W. Wilczynski, and T. Salat, I-convergence, Real Analysis Ex-change 26 (2000), 669–686.

[5] V. Balaz, J. Cervenansky, P. Kostyrko, and T. Salat, I-convergence and I-continuity of real functions, Acta Math., Faculty of Natural Sciences, Constantinethe Philosopher University Nitra 5 (2002), 43–50.

[6] B. Hazarika, On ideal convergence in topological groups, Scientia Magna, Interna-tional Book Series, 7 (2011), 42–48.

Faculty of Science, Department of Mathematics, Dokuz Eylul University, Izmir, TurkeyE-mail: [email protected]

45


THE BISHOP–PHELPS–BOLLOBÁS THEOREMFOR OPERATORS,AND LINDENSTRAUSS PROPERTIES A AND B

Richard M. Aron

Let X be a Banach space with dual X∗. This expository talk will review the Bishop–Phelps Theorem (1961) on density inX∗ of those continuous linear functionals that attaintheir norm, as well as some of the work that this theorem engendered. In particular, wewill discuss Bollobás’ contribution of 1970 as well as Lindenstrauss’ 1963 version for oper-ators. Finally, we will describe recent work relating properties A and B of Lindenstraussto the Bishop–Phelps–Bollobás property for operators. (All terms will be defined.) Thisrecent material is joint work with Yun Sung Choi, Sun Kwang Kim, Han Ju Lee, andMiguel Martín.

REFERENCES

[1] M. D. Acosta, R. M. Aron, D. García, and M. Maestre, The Bishop-Phelps-Bollobás Theorem for operators, J. Funct. Anal. 254 (2008), 2780–2799.

[2] R. M. Aron, Y. S. Choi, S. K. Kim, H. J. Lee, and M. Martín, The Bishop-Phelps-Bollobás version of Lindenstrauss properties A and B, preprint.

[3] E. Bishop and R. R. Phelps, A proof that every Banach space is subreflexive,Bull. Amer. Math. Soc. 67 (1961), 97–98.

[4] B. Bollobás, An extension to the theorem of Bishop and Phelps, Bull. London.Math. Soc. 2 (1970), 181–182.

[5] J. Lindenstrauss, On operators which attain their norm, Israel J. Math. 1 (1963),139–148.

Department of Mathematics, Kent State University, Kent, OH 44242, USAE-mail: [email protected]

46


QUANTITATIVE GROTHENDIECK PROPERTY

Hana Bendová

A Banach space X is Grothendieck if the weak and the weak∗ convergences of se-quences in the dual space X∗ coincide. The space ℓ∞ is a classical example of aGrothendieck space due to Grothendieck. We introduce a quantitative version of theGrothendieck property, we prove a quantitative version of the above-metioned Grothendieck’sresult and we construct a Grothendieck space which is not quantitatively Grothendieck.

Joint work with Ondřej Kalenda.

Department of Mathematical Analysis, Faculty of Mathematics and Physics, Charles Uni-versity in Prague, Sokolovská 83, 186 75 Praha 8, Czech RepublicE-mail: [email protected]

47


HAHN SPACES IN FRÉCHET SPACES

Johann Boos

For real sequence spaces E Bennett, Boos and Leiger introduced in [3] the notion ofa Hahn space: E is called a Hahn space if

χ ∩ E ⊂ F =⇒ E ⊂ F for any FK-space F ,

where χ denotes the set of all sequences of 0′s and 1′s. In that case we say that E hasthe Hahn property. One of the main results tells us that an FK-space E is a Hahn spaceif and only if the linear hull 〈χ ∩ E〉 of χ ∩ E is dense and barrelled in E. Therefore wecan use the Hahn property of FK-spaces to state barrelledness of certain sequence spaceswhich in general is a non-trivial problem.

The aim of this talk is to present a new concept for the introduction and investigationof the notion of a Hahn space in the case of subspaces of Fréchet spaces. The talk ismainly based on the “Diplomarbeit” (master thesis) of Lothar Komp and on a joint paperwith him (cf. [1, 2]).

REFERENCES

[1] L. Komp, Hahn-Eigenschaften von Folgenräumen, Diplomarbeit (Master Thesis),FernUniversität in Hagen, 2009.

[2] L. Komp and J. Boos, Hahn spaces in Fréchet spaces and applications to realsequence spaces, Functiones et Approximatio (accepted 2013, to appear 2014).

[3] G. Bennett, J. Boos, and T. Leiger, Sequences of 0’s and 1’s, Studia Math.149 (2002), 75–99.

[4] G. Bennett and N. J. Kalton, Inclusion theorems for K-spaces, Canad. J. Math.25 (1973), 511–524 .

[5] H. Hahn, Über Folgen linearer Operationen, Monatsh. Math. 32 (1922), 3–88.

⋆ A series of other papers coauthored by M. Parameswaran or Toivo Leiger and/orMaria Zeltser.

Fakultät für Mathematik und Informatik, FernUniversität in Hagen, 58084 Hagen, Ger-manyE-mail: [email protected]

48


HARDY INEQUALITIES ANDINTERPOLATION OF LORENTZ SPACESASSOCIATED TO A VECTOR MEASURE (I)

Ricardo del Campo, Antonio Fernández, Antonio Manzano, Fer-nando Mayoral, and Francisco Naranjo

It is well known that if the classical Lions–Peetre real interpolation method (·, ·)θ,q(0 < θ < 1 ≤, q ≤ ∞) is applied to a pair (X,L∞(µ)), the result is the Lorentz spaceLp,q(µ) with p = 1

1−θ , for every quasi-Banach space X such that L1(µ) ⊆ X ⊆ L1,∞(µ)and any scalar positive measure µ. See, e.g., [2].

The aim of this communication is to extend this result in a twofold direction: fromscalar measures µ to vector measures m, and from the classical real interpolation method(·, ·)θ,q to general real interpolation methods (·, ·)ρ,q associated to parameter functions ρ.For parameter functions ρ in certain classes of functions, these spaces (X0, X1)ρ,q werestudied, first by Kalugina [4] and Gustavsson [3], and later by Persson [5] and otherauthors.

This extension procedure carries the necessity of introducing suitable Lorentz spacesΛqv(‖m‖) associated to a vector measure m and a weight v which fit with our interpolationspaces. As a consequence of our interpolation results, we will find conditions under whichsuch spaces are actually normable quasi-Banach spaces.

Our approach is based on the relationship of the pair (ρ, q) with the Ariño–Muckenhouptweights (see [1] and [6]), and sheds light even to the scalar measure case, providing adifferent point of view for it.

REFERENCES

[1] M. A. Ariño and B. Muckenhoupt, Maximal functions on classical Lorentzspaces and Hardy’s inequality with weights for nonincreasing functions, Trans. Amer.Math. Soc. 320 (1990), 727–735.

[2] J. Bergh and J. Löfström, Interpolation Spaces, An Introduction, Springer,Berlin, 1976.

[3] J. Gustavsson, A function parameter in connection with interpolation of Banachspaces, Math. Scand. 42 (1978), 289–305.

[4] T. F. Kalugina, Interpolation of Banach spaces with a functional parameter, Re-iteration theorem (Russian, with English summary), Vestnik Moskov. Univ. Ser. IMat. Meh. 30 (1975), 68–77.

49

[5] L. E. Persson, Interpolation with a parameter function, Math. Scand. 59 (1986),199–222.

[6] E. Sawyer, Boundedness of classical operators on classical Lorentz spaces, StudiaMath. 96 (1990), 145–158.

Escuela Técnica Superior de Ingeniería Agronómica, Universidad de Sevilla, Carreterade Utrera, km 1, 41013, Sevilla (Spain)E-mail: [email protected] Técnica Superior de Ingenieros, Universidad de Sevilla, Camino de los des-cubrimientos, 41092, Sevilla (Spain)E-mails: [email protected], [email protected], [email protected] Politécnica Superior, Universidad de Burgos, C/ Villadiego, 09001, Burgos (Spain)E-mail: [email protected]

50


AN EXTENDED TAUBERIAN THEOREMFOR THE WEIGHTED MEAN METHODOF SUMMABILITY

İbrahim Çanak and Ümit Totur

We prove a new Tauberian-like theorem to retrieve slow oscillation of a real sequenceout of the weighted mean summability of its generator sequence and some conditions.

REFERENCES

[1] İ. Çanak, An extended Tauberian theorem for the (C, 1) summability method, Appl.Math. Lett. 21 (2008), 74–80.

[2] İ. Çanak and Ü. Totur, Some Tauberian theorems for the weighted mean methodsof summability, Comput. Math. Appl. 62 (2011), 2609–2615.

[3] İ. Çanak and Ü. Totur, Some Tauberian theorems for Borel summability methods,Appl. Math. Lett. 23 (2010), 302–305.

[4] İ. Çanak and Ü. Totur, A condition under which slow oscillation of a sequencefollows from Cesàro summability of its generator sequence, Appl. Math. Comput.216 (2010), 1618–1623.

[5] İ. Çanak, Ü. Totur, and B. P. Allahverdiev, Tauberian conditions with con-trolled oscillatory behavior, Appl. Math. Lett. 25 (2012), 252–256.

[6] İ. Çanak, Ü. Totur, and M. Dik, One-sided Tauberian conditions for (A, k)summability method, Math. Comput. Modelling 51 (2010), 425–430.

[7] F. Móricz and B. E. Rhoades, Necessary and sufficient Tauberian conditionsfor certain weighted mean methods of summability, Acta Math. Hungar. 66 (1995),105–111.

[8] H. Tietz, Schmidtsche Umkehrbedingungen für Potenzreihenverfahren, Acta Sci.Math. 54 (1990), 355–365.

[9] H. Tietz and K. Zeller, Tauber-Bedingungen für Verfahren mit Abschnittskon-vergenz, Acta Math. Hungar. 81 (1998), 241–247.

51

Department of Mathematics, Ege University, 35100 İzmir, TurkeyE-mail: [email protected] of Mathematics, Adnan Menderes University, 09010 Aydın, TurkeyE-mail: [email protected]

52


RADON–NIKODÝM INDEXES AND MEASURESOF WEAK NONCOMPACTNESS

Bernardo Cascales

We introduce the index of representability of a vector measure and the index ofdentability of subsets of Banach spaces. We study the relationship between them aswell as with other well-known indexes such as those that give the measure of non weak-compactness of bounded sets. We establish a number of inequalities that summarize,sharpen and offer new ways to some aspects of the theory of Radon–Nikodým propertyin Banach spaces previously studied by many authors. We offer examples that provethat our inequalities are sharp. A tool for the above is the concept of ultrafilter andthe existence of liftings in complete probability spaces. In the first part of the lecturewe recall the concepts of filter and ultrafilter, a little bit of their history and we givereferences to some nice applications in algebra and analysis.

This lecture covers general results as well as results from a recent paper with AntonioPérez and Matias Raja. This research was partially supported by the Spanish MCI andFEDER project MTM2011-25377

Departamento de Matemáticas, Murcia, SpainE-mail: [email protected]

53


COMMUTATIVE SUBALGEBRAS OF THE ALGEBRAOF SMOOTH OPERATORS

Tomasz Ciaś

In this talk we deal with the noncommutative Fréchet ∗-algebra L(s′, s) (so-calledalgebra of smooth operators) of continuous linear operators from the space

s′ :=ξ = (ξj)j∈N ∈ CN : ∃q ∈ N0 |ξ|′q :=

( ∞∑

j=1

|ξj |2j−2q

) 12

<∞

of slowly increasing sequences to the space

s :=

ξ = (ξj)j∈N ∈ CN : ∀q ∈ N0 |ξ|q :=

( ∞∑

j=1

|ξj |2j2q) 1

2

<∞

of rapidly decreasing sequences, where multiplication and involution are defined, in a nat-ural way, through the composition of operators on the Hilbert space ℓ2 and the hilbertianinvolution. The algebra L(s′, s) can be represented as the algebra of matrices (aj,k)j,k∈Nsuch that supj,k∈N |aj,k|jqkq < ∞ for all q ∈ N0 (so-called rapidly decreasing matriceswith matrix multiplication and matrix complex involution) and as the algebra S(R2) ofSchwartz functions on R2 with the Volterra convolution (f ·g)(x, y) :=

∫R f(x, z)g(z, y)dz

as multiplication and involution f∗(x, y) := f(y, x). In these forms, the algebra L(s′, s)usually appears and plays a significant role in papers on K-theory of Fréchet algebras,C∗-dynamical systems and in noncommutative geometry (see e.g. papers of Cuntz, Black-adar, Elliot, Natsume, Nest, Phillips).

We show that every closed commutative ∗-subalgebra of L(s′, s) is generated by asingle (normal) operator and has a Schauder basis consisting of pairwise orthogonalprojections. As a by-product we get a Hölder continuous functional calculus in L(s′, s).Next we show that every closed commutative ∗-subalgebra of L(s′, s) which is isomorphic(as a Fréchet space) to some complemented subspace of the space s is already isomorphic(as a Fréchet ∗-algebra) to some subalgebra of the algebra s (pointwise multiplication andinvolution). We also give an example of a closed commutative ∗-subalgebra of L(s′, s)which is not isomorphic to any subalgebra of s.

REFERENCES

[1] Tomasz Ciaś, On the algebra of smooth operators. Preprint available athttp://arxiv.org/abs/1304.7189, submitted to a journal.

54

Faculty of Mathematics and Computer Science, Adam Mickiewicz University in Poznań,Umultowska 87, 61-614 Poznań, PolandE-mail: [email protected]

55


ON THE HERMITIAN PART OF RICKART *-RINGS

Janis Cırulis

An associative involution ring R is said to be Rickart if the right annihilator of everyelement of R is generated by a projection (necessarily unique). The most importantexample of such a ring is the ring B(H) of bounded linear operators of a Hilbert spaceH. The Hermitian part S of R consists, by definition, of the self-adjoint elements ofR. If S, like the subset S(H) of self-adjoint Hilbert space operators, does not containnilpotent elements distinct from 0, then the relation on S defined by

x y if and only if x(y − x) = 0

is an order relation (introduced, under the same restriction, for commutative and associa-tive rings by A. Abian in 70ies and later extended to wide classes of arbitrary rings). OnS(H), this order has been studied, exploring also some theory of Hilbert spaces, in [1–3].

In the talk, we give an abstract (axiomatic) description of the partial rings arising asa Hermitian part of a Rickart *-ring, and describe their Abian order structure.

REFERENCES

[1] J. Cırulis, Further remarks on an order for quantum observables, Math. Slovaca63 (2013), to appear.

[2] S. Gudder, An order for quantum observables, Math. Slovaca 56 (2006), 573–589.

[3] S. Pulmannová and E. Vinceková, Remarks on the order for quantum observ-ables, Math. Slovaca 57 (2007), 589–600.

Institute of Mathematics and Informatics, University of Latvia, 29 Raina b., Riga LV-1459, LatviaE-mail: [email protected]

56


MULTI-NORMS

Harold Garth Dales

Let E be a Banach space. A multi-norm based on E is a sequence

(‖ · ‖n : n ∈ N)

of norms, where ‖ · ‖n is defined on En, to satisfy certain axioms. The theory of multi-normed spaces is related to, but different from, the theory of operator spaces. It hasstrong connections with the theories of absolutely summing operators and of cross prod-ucts on tensor products of spaces.

I shall review the theory, and discuss some of its applications. In particular, I shalldiscuss the equivalence of multi-norms, multi-bounded operators, and applications toBanach lattices.

Details are given in the publications listed below.

REFERENCES

[1] O. Blasco, H. G. Dales, and H. L. Pham, Equivalences involving (p, q)-multi-norms, in preparation.

[2] H. G. Dales and M. E. Polyakov, Multi-normed spaces, Dissertationes Math.488 (2012) 1–165.

[3] H. G. Dales, M. Daws, H. L. Pham, and P. Ramsden, Multi-norms and theinjectivity of Lp(G), J. London Math. Society 86 (2012), 779–809.

[4] H. G. Dales, M. Daws, H. L. Pham, and P. Ramsden, Equivalence of multi-norms, Dissertationes Math., to appear.

University of Lancaster, Lancaster, LA1 4YF, United KingdomE-mail: [email protected]

57


A CHARACTERIZATIONOF OPERATORS WITH p-SUMMING ADJOINTVIA p-LIMITED SETS

Juan Manuel Delgado and Cándido Piñeiro

A subset A of a Banach space X is limited if and only if for every weak∗ null sequence(x∗n) in X∗ there exists a sequence (αn) ∈ c0 such that |〈x∗n, x〉| ≤ αn for all x ∈ A andn ∈ N. As an extension of this notion, a subset A of a Banach space X is said to bep-limited (p ∈ [1,∞)) if for every weakly p-summable sequence (x∗n) in X∗ there exists(αn) ∈ ℓp such that |〈x∗n, x〉| ≤ αn for all x ∈ A and n ∈ N [2]. Some basic propertiesrelated to this notion are showed as well as its connections with (different forms of)compactness. As an application, we give a characterization of operators with p-summingadjoint as those who map relatively compact sets to p-limited sets.

The research was supported by MTM2009-14483-C02-01 and MTM2012-36740-C02-01 projects (Spain).

REFERENCES

[1] J. M. Delgado and C. Piñeiro, A note on p-limited sets, preprint.

[2] A. K. Karn and D. P. Sinha, An operator summability in sequences in Banachspaces, arXiv:1207.3620 [math.FA] (2012).

Higher Technical School of Architecture, University of Seville, 41012 Seville, Av. ReinaMercedes, 2, SpainE-mail: [email protected] of Experimental Sciences, University of Huelva, 21071 Huelva, Campus Univer-sitario de El Carmen, SpainE-mail: [email protected]

58


SUMS

Joe Diestel

One of the earlies results we meet in “advanced calculus” is a theorem about conver-gence of infinite series of real numbers: a series can converge unconditionally (in whichcase, all rearrangements converge to the same sum and the series is, in fact, absolutelyconvergent) or the convergence is conditional (in which case, given any real number thereis a rearrangement of the series that converges to that number). This result has naturalextensions to finitely many dimensions. What about the infinital situation?

The talk is aimed at people from outside analysis. Analysts will, I fear, find the talkpedestrian but we all need a good “walk in the park” occasionally.

Department of Mathematical Sciences, Kent State University, Kent, Ohio 44242, USAE-mail: [email protected]

59


AN OUTER MEASURE ON A COMMUTATIVE RING

Dariusz Dudzik

We propose a construction of an outer measure on a commutative ring R with identity,similar to standard constructions, but using prime ideals as covering sets.

Let P ⊆ Spec(R) be a family of prime ideals such that⋃

P = R \R×,

where R× is the set of invertible elements of R, and let µ be a measure on P. The outermeasure µ∗ : 2R −→ [0,+∞] induced by µ is defined by

µ∗(A) = infS∈Ω(A)

µ(S),

whereΩ(A) =

S ⊆ P : S is µ-measurable,

⋃S ⊇ A \R×

.

In the talk, we will prove several properties of µ∗ and provide a few examples relatedto algebraic geometry, functional analysis and number theory.

This is a joint work with Marcin Skrzyński.

REFERENCES

[1] M. F. Atiyah and I. G. MacDonald, Introduction to Commutative Algebra, West-view Press, 1994.

[2] H. Federer, Geometric Measure Theory, Springer, Berlin-Heidelberg-New York,1969.

[3] W. Rudin, Functional Analysis, McGraw-Hill, 1991.

Institute of Mathematics, Pedagogical University of Cracow, ul. Podchorażych 2, 30-084Kraków, PolandE-mail: [email protected]

60


DIFFERENT KINDS OF INTEGRALSIN THE SAME FORMULA

Antonio Fernández, Fernando Mayoral, and Francisco Naranjo

We study some relationships between the Bartle–Dunford–Schwartz integral of ascalar valued function f, with respect to a vector measure m, and the Dunford, Pet-tis or Bochner integrals of its (vector valued) distribution function mf . The Dunford(or Pettis) integrability of mf is strongly related to the weak integrability (or the in-tegrability) of f in the sense of Bartle–Dunford–Schwartz. In the case of the Bochnerintegrability of mf , a new function space appears. It is defined through the Choquetintegrability of f with respect to the semivariation ‖m‖ of the measure m. We also studythis space and present its main properties.

The research was partially supported by La Junta de Andalucía and the Ministeriode Economía y Competitividad under the projects MTM2009-14483-C02 and MTM2012-36740-C02.

Escuela Técnica Superior de Ingeniería, University of Sevilla, Camino de los Descubrim-ientos, s/n. 41092 Sevilla, SpainE-mails: [email protected], [email protected], [email protected]

61


APPROXIMATION PROPERTIES IN BANACH SPACES

Gilles Godefroy

Most of the classical Banach spaces have bases, and thus good approximation schemeswhich allow to perform explicit computations. However, Per Enflo opened a new era offunctional analysis in 1972 by showing the existence of Banach spaces which fail theapproximation property. Progress has been made since then, but a bunch of naturalproblems on approximation properties remain open, for instance in the newly consideredclass of “free Banach spaces”. We will investigate some of these problems.

Institute of Mathematics of Jussieu, FranceE-mail: [email protected]

62


“ALMOST” CAUCHY PROPERTYFOR THE SEQUENCE OF PARTIAL SUMSOF FOURIER SERIES OF FUNCTIONS IN Lp, p > 1

Igor Bloshanskii and Denis Grafov

Let a 2π-periodic (in each argument) function f ∈ L1(TN ), TN = [−π, π)N , N ≥ 1,be expanded in the multiple trigonometric Fourier series f(x) ∼ ∑

ckeixk, and Sn(x; f),

n ∈ ZN+ , be a rectangular partial sum of this series. And let a function g ∈ L1(RN ),N ≥ 1, be expanded in the multiple Fourier integral g(x) ∼

∫g(ξ)eixξdξ, and Jα(x; g),

α ∈ RN+ , be a proper Fourier integral.Suppose that g(x) = f(x) for x ∈ TN . Let us define by Rα(x; f, g) the difference

Rα(x; f, g) = Sn(x; f) − Jα(x; g), and by Rα(x; f) the difference Rα(x; f) = Sn(x; f) −Jα(x; g) if g(x) = 0 out of TN , where n = [α] = ([α1], . . . , [αN ]) ∈ ZN+ ([t] is the integralpart of t ∈ R1

+). In [1] it was proved that for N = 2 and p > 1 Rα(x; f, g) → 0 as α→ ∞(i.e., min

1≤s≤Nαs → ∞) almost everywhere on T2. In the same paper it was proved that

conditions N = 2, p > 1 are essential. In particular, a function f0 ∈ C(TN ), N ≥ 3,was constructed such that lim

α→∞|Rα(x; f0)| = +∞ everywhere inside TN . The question

arises: how the difference Rα(x; f, g) behaves if the components nj and αj of vectorsn ∈ ZN+ and α ∈ RN+ are connected by relation

| αj − nj |≤ const, j = 1, . . . , N. (1)

The following theorem answers this question for N = 2.

Theorem 1. For any α = (α1, α2), α ∈ R2+, satisfying condition (1), and for any

functions g(x) and f(x) such that g ∈ Lp(R2), f ∈ Lp(T2), p > 1, and g(x) = f(x) forx ∈ T2,

limα1, α2→∞

Rα1, α2(x; f, g) = 0 almost everywhere on T2.

Further, let us define RSn+m(x; f) = Sn+m(x; f)−Sn(x; f), n, m ∈ ZN+ . The follow-ing result is equivalent to Theorem 1.

Theorem 2. For any bounded sequence m(n), m(n) ∈ Z2+, n ∈ Z2

+, and for anyfunction f ∈ Lp(T2), p > 1,

limn→∞

RSn+m(n)(x; f) = 0 almost everywhere on T2.

Note that this estimate is true for the divergent a.e. Fourier series as well.As we have said above, for N ≥ 3 Theorem 1 is not true even in C(TN ). It is not

difficult to prove that the assertion of Theorem 2 is either not true in this class. In this

63

case, the following question arises: how the differences Rα(x; f, g) and RSn+m(x; f) inthe classes Lp, for N ≥ 3, behave if some of the components nj of the vector n = [α]are elements of (single) lacunary sequences (k(s), k(s) ∈ Z1

+, is a lacunary sequence ifk(s+1)/k(s) ≥ q > 1, s = 1, 2, . . . ).

The possibility to obtain new results in the case of additional restrictions on the vectorn = [α] is connected with the fact that in the classes Lp, p > 1, the “lacunary” subse-quences of partial sums of multiple Fourier series have better properties of convergencea.e. in comparison with the whole sequence Sn(x; f).

The partial answers on the latter question are the following results (which, in particu-lar, show that for N ≥ 3 the differences Rα(x; f, g) and RSn+m(x; f) are not equivalent).

Theorem 3. There exists a function f ∈ C(TN ), N ≥ 3, such that for any sequenceα = (α3, . . . , αN ) ∈ RN−2

+

limn1, n2, α→∞

|Rn1, n2, α(x; f)| = +∞ everywhere inside TN .

Theorem 4. For any bounded sequence m(n), m(n) ∈ Z2+, n = (n1, n2) ∈ Z2

+, for anylacunary sequences n(λj)j , n(λj)j ∈ Z1

+, λj = 1, 2, . . . , j = 3, . . . , N, and for any functionf ∈ Lp(TN ), p > 1, N ≥ 3, almost everywhere on TN

limn1, n2, λ3, ...,λN→∞

RSn1+m1(n), n2+m2(n), n

(λ3)3 , ...,n

(λN )

N

(x; f) = 0.

The work is supported by Russian Foundation for Basic Research (Grant 11-01-00321).

REFERENCES

[1] I. L. Bloshanskii, On equiconvergence of expansions in multiple trigonometricFourier series and Fourier integral, Matem. Zametki 18 (1975), 153–168.

Department of Mathematical Analysis and Geometry, Moscow State Regional University,105005 Moscow, 10a Radio Street, RussiaE-mails: [email protected], [email protected]

64


ON BLOCKS OF SKELETON TOLERANCES

Joanna Grygiel and Katarzyna Grygiel

A tolerance relation of a lattice L is a reflexive and symmetric relation compatiblewith operations of the lattice. Let T be a tolerance on a lattice L and let ∅ 6= X ⊆ L. Ifevery two elements of X are in the relation T , then we call X a preblock of T . Blocks ofT are maximal preblocks (with respect to inclusion). A tolerance T of a finite lattice Lis called a glued tolerance if its transitive closure is the total relation L2. The (unique)smallest glued tolerance of L is called the skeleton tolerance of L.

It is well known that every finite lattice is a skeleton of a finite distributive lattice [3]or, even more, of infinitely many finite distributive lattices [2]. However, we are goingto show that there are lattices which cannot be blocks of the skeleton tolerance of anyfinite lattice. Of course, it is clear for distributive or modular lattices, especially, as it isknown that their blocks of the skeleton tolerance are maximal boolean or, respectively,complemented (or, equivalently, atomistic) intervals of such lattices [1]. Although in themodular case blocks of the skeleton tolerance are H-irreducible (see, e.g., [2]), it is nottrue in the general case.

In the talk we provide a full characterization of lattices which can be blocks of theskeleton tolerance relation of a finite lattice. Moreover, we formulate a necessary condi-tion for a lattice to be such a block in the case of finite distributive lattices with at mostk-dimensional maximal boolean intervals.

This research was supported by the National Science Center of Poland, grant number2011/01/B/HS1/00944.

Joint work with Anetta Górnicka.

REFERENCES

[1] A. Day and Ch. Herrmann, Gluings of modular lattices, Order 5 (1988), 85–101.

[2] J. Grygiel, The Concept of Gluing for Lattices, Wyższa Szkoła Pedagogiczna wCzęstochowie, Częstochowa, 2004.

[3] Ch. Herrmann, S-verklebte Summen von Verbänden, Math. Z. 130 (1973), 255–274.

Jan Dlugosz University in Czestochowa, al. Armii Krajowej 13/15, 42-218 Czestochowa,PolandE-mail: [email protected] University in Krakow, ul. Prof. S. Łojasiewicza 6, 30-348 KrakówE-mail: [email protected]

65


THE SZLENK INDEX OF Lp(X)

Petr Hájek and Thomas Schlumprecht

We find an optimal upper bound on the values of the weak∗-dentability index Dz(X)in terms of the Szlenk index Sz(X) of a Banach space X with separable dual. Namely,if Sz(X) = ωα, for some α < ω1, and p ∈ (1,∞), then

Sz(X) ≤ Dz(X) ≤ Sz(Lp(X)) ≤ωα+1 if α is a finite ordinal,ωα if α is an infinite ordinal.

The first author’s research was supported in part by GAČR P201/11/0345, ProjectBarrande 7AMB12FR003, and RVO: 67985840. The second author’s research was par-tially supported by NSF grants DMS0856148 and DMS1160633.

Mathematical Institute, Czech Academy of Science, Žitná 25, 115 67 Praha 1, CzechRepublic; Department of Mathematics, Faculty of Electrical Engineering, Czech TechnicalUniversity in Prague, Zikova 4, 160 00, PragueE-mail: [email protected] of Mathematics, Texas A&M University, College Station, TX 77843; De-partment of Mathematics, Faculty of Electrical Engineering, Czech Technical Universityin Prague, Zikova 4, 160 00, PragueE-mail: [email protected]

66


STRUCTURE OF GENERATING SETS FOR REVERSIBLECOMPUTATIONS

Jaak Henno

The IBM researcher Rolf Landauer stated in 1961 a law [1], which connects compu-tations with physics (thermodynamics): when a computational system erases a bit ofinformation, it must dissipate ln(2) ∗ kT energy (heat); here k is Boltzmann’s constantand T is the temperature; currently this law is referred as the Landauer’s principle,but the idea of equivalence of information and thermodynamic entropy was consideredalready, e.g., by Szilard [2].

In order not to dissipate heat, computation should not erase anything. Such a compu-tation is reversible – every step of the computation can be done also in backwards (undo).A reversible computation does not generate heat and according to current knowledge itcan be implemented on quantum level, using qubits instead of ordinary bits. But re-versible computations/functions occur also in ordinary computations – cryptographicfunctions, many image-editing functions (lossless image compression) etc. all should bereversible. A quantum computer could execute many currently difficult computationaltasks much quicker, e.g., a quantum algorithm can solve the integer factorization problemexponentially faster than the best-known classical algorithms [3]. Thus, in recent yearsreversible computations have become a very important research topic for its enormouspossibilities in low power CMOS design, quantum computing and nanotechnology.

The binary Boolean functions &,∨,→,↔,⊕,NAND etc. are not reversible, but nega-tion is.

However, every computation can be embedded into a reversible one – every Turing ma-chine can be made reversible [4]. Among several constructs for converting non-reversiblefunctions into reversible ones, the most often is cited the Toffoli construction [5] (“Toffoligate”) – a ternary Boolean function with ternary output (a reversible function shouldhave the same number of inputs-outputs), where result of conjunction x1&x2 of the firsttwo arguments is used to flip the state of the third argument, i.e., it implements the 3-aryreversible function (x1, x2, x3) 7→ (x1, x2, (x1&x2)⊕x3). Usually reversible computationsare considered only for binary, i.e., Boolean logic, but, e.g., life encodes its programs(genes) in a 4-valued logic.

For “real” computing one of the most essential problems are bases – sets of functionsused to express (calculate) every other function; functions of a base are implementedeither in hardware (processor) or using processor functions (compiler). Every n-aryreversible function of m-valued logic implements a substitution on the set of mn n-placevectors (inputs to outputs) whose coordinates are from the set 0, 1, . . . ,m − 1, i.e.,is an element of the group Smn . Generating sets for the symmetric substitution groupshave been studied extensively for quite a time (see, e.g., [6, 7]), but for “real-world” the

67

implementations derived from algebraic group-theoretic properties of the group Smn maynot be always what is the best/cheapest to implement.

Problems of cost of implementation and usability of bases in processors (implementingnon-reversible computations) have been studied for quite a long time. The Toffoli idea(storing intermediate values fliping the state of some wire) has been used to show that if aBoolean function can be embedded into an even permutation with polynomial-size cyclerepresentation, then the function can be implemented by a polynomial-size reversiblecircuit [8].

Here it will be shown that the Toffoli idea can be used to convert every generatingset of functions (base) in m-valued logic into generating set (base) of reversible functionsof m-valued logic; the idea is similar to what has been used in [9]. This constructionpreserves partial order of bases [10], based on (minimal) depth dF (f) of implementationof function f in base F :

F16dF2

if and only if there exists a constant k such that for every function f of m-valued logic

dF1(f) 6 dF2(f) + k;

for binary Boolean bases k = 2.

REFERENCES

[1] R. Landauer, Information is Inevitably Physical, in Feynman and Computation,ed. A. J. G. Hey, Addison Wesley Longman, Reading MA, 1998.

[2] L. Szilard, On the Decrease of Entropy in a Thermodynamic System by the In-tervention of Intelligent Beings, Zeitschrift fur Physik 53 (1929) 840–856. Englishtranslation in The Collected Works of Leo Szilard: Scientific Papers, B. T. Feld andG. Weiss Szilard (eds.), Cambridge, Massachusetts, MIT Press, 1972, 103–129.

[3] P. Shor, Polynomial-time algorithms for prime factorization and discrete loga-rithms on a quantum computer, SIAM J. Computing, 26 (2005) 1484–1509.

[4] C. H. Bennett, Logical reversibility of computation, IBM J. Research and Devel-opment, 17 (1973), 525–532.

[5] T. Toffoli, Reversible Computing, Tech. Memo MIT/LCS/TM-151, MIT Lab forCS, 1980.

[6] G. A. Miller, Theory and applications of finite groups, HF Blichfeldt, IE DICK-SON – 1916.

[7] G. A. Miller, Possible orders of two generators of the alternating and of thesymmetric group, Bull. Amer. Math. Soc., 7 (1901), 424.URL: http://www.ams.org/journals/tran/1928-030-01/S0002-9947-1928-1501419-9/S0002-9947-1928-1501419-9.pdf

68

[8] A. Brodsky, Reversible Circuit Realizations of Boolean Functions, Proc. 3rd IFIPInt. Conf. Theor. Computer Sc., 2004.

[9] A. De Vos, B. Desoete, F. Janiak, and A. Nogawski, Control gates for re-versible computers, Proc. 11th Int. Workshop on Power and Timing Modeling, Op-timization and Simulation, Yverdon, Sept. 2001, 9.2.1–9.2.10.

[10] J. Henno, On equivalent sets of functions, Discrete Appl. Math., 4 (1982), 153–156.

Tallinn University of Technology, EstoniaE-mail: [email protected]

69


APPROXIMATIONIN THE WEIGHTED LEBESGUE SPACES

Daniyal Israfılov

Let Lp (Γ) and Ep (G) be the set of all measurable complex valued functions suchthat | f |p is Lebesgue integrable on Γ with respect to the arclength and the Smirnovclass of analytic functions in G, respectively.

For p > 1, Lp (Γ) and Ep (G) are Banach spaces with respect to the norm

‖ f ‖Ep(G):=‖ f ‖Lp(Γ):=

∫

Γ

| f (z) |p| dz |

1/p

.

By υ we denote a weight function on Γ, i.e., υ : Γ → [0,∞] , for which the setυ−1 (0,∞) has measure zero.

We assume that υ ∈ Ap (Γ), i.e., satisfies the well-known Muckenhout condition onΓ.

With every weight υ ∈ Ap (Γ), 1 < p <∞, we associate the weighted Lebesgue spacesLp (Γ, υ), consisting of all measurable functions f on Γ such that

‖ f ‖Lp(Γ,υ):=

∫

Γ

| f (z) |p υ (z) | dz |

1p

<∞

and the weighted Smirnov spaces Ep (G, υ) with

‖ f ‖Ep(G,υ):=f ∈ E1 (G) : ‖ f ‖Lp(Γ,υ)<∞

.

The spaces Lp (Γ, υ) and Ep (G, υ) become Banach spaces if υ ∈ Ap (Γ).In this talk we discuss the direct problems of approximation theory by rational func-

tions and by polynomials in the spaces Lp (Γ, υ) and Ep (G, υ), respectively. Some specialcases of this problem were investigated in [1] and [2].

REFERENCES

[1] D. M. Israfilov, Approximation by p-Faber polynomials in the weighted Smirnovclass Ep (G,ω) and the Bieberbach polynomials, Constr. Approx., 17 (2001), 335–351.

[2] D. M. Israfilov, Approximation by p-Faber–Laurent rational functions in theweighted Lebesgue spaces, Czechoslovak Math. J., 54 (2004), 751–765.

70

Department of Mathematics Faculty of Sciences and Arts, University of Balikesir, 10145Balikesir TurkeyE-mail: [email protected]

71


NICE CATEGORIES

George Janelidze

General mathematical structures, e.g., in the sense of Bourbaki, can be used as apossible motivation for introducing categories and for categorical unification of mathe-matics. The unification then leads to a new way of thinking, not in terms of manipulatingwith elements of mathematical structures, but in terms of composing morphisms betweenobjects of an abstract category, creating a new algebra that applies to all areas of mathe-matics. However, many categorical concepts and arguments require additional conditionson a ground category, which, in more than sixty years of development, created a verysophisticated hierarchy of “nice” categories. One of lines in this hierarchy, which we shallessentially follow trying to make our presentation self-contained, can be briefly describedas: finitely complete–regular–exact–semi-abelian–abelian. The conditions involved hereare perfectly well understood in many classical-algebraic examples, less well in topologyand topological algebra, and much less in functional analysis. At the end of the talksome recent results and open problems will be mentioned.

The research was partially supported by South African NRF.

Department of Mathematics and Applied Mathematics, University of Cape Town, 7701Rondebosch, South AfricaE-mail: [email protected]

72


REGULAR METHOD OF SUMMABILITYAND THE WEAK FATOU PROPERTY

Eduardo Jiménez Fernández, María Aranzazu Juan, and EnriqueA. Sánchez-Pérez

Let X(µ) be a σ-order continuous Banach function space where µ is a finite measure.In this talk we characterize the regular methods of summability in terms of a weakσ-Fatou properties of these spaces. We generalize this result to a more general classof Banach function spaces, in particular to spaces of classes of measurables integrablefunctions with respect to a vector measure ν on a δ-ring.

The authors gratefully acknowledge the support of the Ministerio de Economía y Com-petitividad, project #MTM2012-36740-C02-02, and Junta de Andalucía and FEDER,grant #P09-FQM-4911.

Universidad de Granada, Av de la Fuente Nueva Granada, SpainE-mail: [email protected] Politecnica de Valencia, Camino de Vera s/n. 46022 Valencia. SpainE-mail: [email protected]

73


UNCONDITIONAL IDEALS OF COMPACT OPERATORS

Marje Johanson

The subspace K(X,Y ) of compact operators from a Banach space X to a Banachspace Y is called an ideal in the Banach space L(X,Y ) of all bounded linear operatorsif there exists a norm one projection P on L(X,Y )∗ with kerP = K(X,Y )⊥. Moreover,if the projection P satisfies ‖IL(X,Y )∗ − 2P‖ = 1, where IL(X,Y )∗ denotes the identityoperator of L(X,Y )∗, then K(X,Y ) is a u-ideal (or in longer, unconditional ideal) inL(X,Y ).

In [CK] P. G. Casazza and N. J. Kalton prove the following result.

Theorem. Let X be a separable reflexive Banach space with the approximation property.Then K(X,X) is a u-ideal in L(X,X) if and only if X has the unconditional metricapproximation property.

We generalize the result and view different criteria for compact operators to form u-idealsin the space of continuous linear operators. Also, we show that u-ideals of compactoperators are separably determined.


REFERENCES

[CK] P. G. Casazza and N. J. Kalton, Notes on approximation properties in sep-arable Banach spaces, in: Geometry of Banach Spaces, Proc. Conf. Strobl (1989)(P. F. X. Müller and W. Schachermayer, eds.), London Math. Soc. Lecture NoteSeries, vol. 158, Cambridge Univ. Press, 1990, pp. 49–63.

[GKS] G. Godefroy, N. J. Kalton, and P. D. Saphar, Unconditional ideals inBanach spaces, Studia Math. 104 (1993), 13–59.

Faculty of Mathematics and Computer Science, University of Tartu, 50409 Tartu, J. Liivi2, EstoniaE-mail: [email protected]

74


APPROXIMATION PROPERTIESOF A BANACH SPACE AND ITS SUBSPACES

William Johnson

1. HAPpy Banach spaces.2. Joint AP for a Banach space X and its subspace Y .3. Erdős meets Lidskii.

Department of Mathematics, Texas A&M Unversity, College Station, TX 77843, USAE-mail: [email protected]

75


COMPATIBLE FUNCTION EXTENSIONAND MAJORITY FUNCTIONS

Kalle Kaarli

Let A be a set, s ⊆ A2, D ⊆ An and f : D → A. We say that f preserves s iffor every 〈x1, y1〉, . . . , 〈xn, yn〉 ∈ s, whenever 〈x1, . . . , xn〉, 〈y1, . . . , yn〉 ∈ D, then also〈f(x1, . . . , xn), f(y1, . . . , yn)〉 ∈ s. We say that f is S-compatible where S is a set ofsubsets of A2 if it preserves all s ∈ S.

Recall that a ternary function f on a set A is called a majority function if f(a, a, b) =f(a, b, a) = f(b, a, a) = a for all a, b ∈ A. The f is said to be a Pixley function, iff(a, b, b) = f(a, b, a) = f(b, b, a) = a for all a, b ∈ A.

In [1] we proved, in particular, the following result.

Theorem. Let A be a finite set and S be a sublattice of the lattice of all equivalencerelations on A. Assume that S contains both the diagonal ∆A and ∇A = A2. Then thefollowing are equivalent:

1. for any positive integer n and any D ⊆ An, every S-compatible function f : D → Acan be extended to an S-compatible function An → A;

2. there exists an S-compatible Pixley function on A.

In the present work the following analog of this theorem has been obtained.

Theorem. Let A be a finite set and S be a set of binary relations on A. Assume that Sis closed with respect to forming intersections, relational products and converse relations,and contains both ∆A and ∇A. Then the following are equivalent:

1. for any positive integer n and any D ⊆ An, every S-compatible function f : D → Acan be extended to an S-compatible function An → A;

2. there exists an S-compatible majority function on A.

REFERENCES

[1] K. Kaarli, Compatible function extension property, Algebra Universalis, 17 (1983),200–207.

Institute of Mathematics, University of Tartu, 50090 Tartu, EstoniaE-mail: [email protected]

76


FINITELY BASED FINITE ALGEBRAS

Keith A. Kearnes

I will discuss the problem of determining whether a finite algebra has a finite basisfor its equational laws.

Department of Mathematics, University of Colorado, Boulder CO 80309-0395E-mail: [email protected]

77


ON SUMMABILITY OF ORTHOGONAL EXPANSIONSAND SHANNON SAMPLING SERIES

Andi Kivinukk

Let X be a fixed Banach space, and we assume that any f ∈ X is in some senserepresentable via an orthogonal expansion f ∼ ∑∞

k=0 Pkf, using a sequence of mutuallyorthogonal projections Pk∞k=0. Often the convergence of that series is achieved usingthe summability method (called the (Φ, λ) method in [1])

U(t)f =∞∑

k=0

ϕ(λk/t)Pkf, t > 0. (1)

If ϕ(u) = (1 − ur)+, r ∈ N, then we get the Zygmund method, denoted by Zrn, n ∈N. Often it forms an approximation process with known order of approximation, i.e.,‖f −Zrnf‖ ≤MrΩr(f, 1/n) → 0, n→ ∞, where Ωr(f, 1/n) is a modulus of continuity, aK-functional or another known sequence.

We will demonstrate that some quite general operators (1) can be considered usinga subordination equality

f − Unf =∞∑

j=r

cj(f − Zrnf), (2)

under the essential assumption ϕ(u) = 1 − ∑∞j=r cju

j . We apply (2) to the Shannonsampling series [2].

This research was partially supported by Estonian Science Foundation Grant 8627.

REFERENCES

[1] G. Kangro, Theory of summability of sequences and series (in Russian), Math-ematical analysis. Akad. Nauk SSSR Vsesojuz. Inst. Naučn. i Tehn. Informacii,Moscow 12 (1974), 5–70.

[2] A. Kivinukk and G. Tamberg, Approximation by Shannon sampling operators interms of an averaged modulus of smoothness, Proc. SampTA, Bremen, July 1–5,2013. (To be published by EURASIP.)

Department of Mathematics, Tallinn University, 10120 Tallinn, 25 Narva Road, EstoniaE-mail: [email protected]

78


TERM-EQUIVALENCE OF SEMILATTICES

Oleg Košik and Peter Mayr

In this talk we will discuss term-equivalence of semilattices.

Institute of Mathematics, University of Tartu, EstoniaE-mail: [email protected] for Algebra, JKU Linz, AustriaE-mail: [email protected]

79


ON STRONG AVERAGESOF SPHERICAL FOURIER SUMS

Olga I. Kuznetsova and Anatolii N. Podkorytov

We study two similar summation methods for multiple Fourier series which areanalogs of the classical strong summation method introduced by Hardy and Littlewood [1]in the one-dimensional case.

Let

SR(f, x) =∑

‖k‖6Rf(k) eik·x, where f(k) =

1

(2π)m

∫

Tm

f(u) e−ik·udu for k ∈ Zm,

be a spherical partial sum for the Fourier series of a function f which is integrable onthe cube Tm = [−π, π]m. We say that the sequence Sn(f, x) is Hp-summable (in whatfollows, p > 1 is a fixed parameter) to the sum S if

1

n

n−1∑

j=0

|Sj(f, x)− S|p −→n→+∞

0.

Similarly, Hp-summability means that

1

R

∫ R

0|Sr(f, x)− S|pdr −→

R→+∞0.

In the one-dimensional case, these methods coincide. Moreover, as stated by Hardy andLittlewood, for a function f its continuity at a point x ∈ T implies its Fourier seriessummability in the above sense.

The question about the Fourier series summability by the Hp and Hp methods in thecase of continuous functions in several variables is mainly related to the behavior of thenorms

Hn,p = sup|f |61

(1

n

n−1∑

j=0

|Sj(f, 0)|p) 1

p

and HR,p = sup|f |61

(1

R

∫ R

0|Sr(f, 0)|pdr

) 1p

.

In the one-dimensional case, they are bounded, which is equivalent to the Hardy-Littlewood’s theorem. In the multidimensional case (m > 1) the situation is different:for any p > 1 the norms Hn,p and HR,p are unbounded [2, 3]. An interesting question isto get an estimation exact in order [4]. Clearly, estimating HR,p one can regard R to bean integer.

80

We show that the norms Hn,p and Hn,p have the same order. More exactly,1

Hn,p ≍ Hn,p ≍

nm−1

2−min 1

2, 1p if m > 3, p > 1;

n12− 1

p min1pln(n+ 1), 1

p−2

if m = 2, p > 2;√

ln(n+ 1) if m = 2, p ∈ [1, 2].

Using standard reasoning, one deduces from these estimations conditions imposed onthe continuous function f that ensure the uniform Hp and Hp-summability of the Fourierseries, i.e., the uniform (relative to x ∈ Rm) convergence to zero of the quantities

1

n

n−1∑

j=0

|Sj(f, x)− f(x)|p and1

R

∫ R

0|Sr(f, x)− f(x)|pdr.

This condition is better formulated in terms of the notion of the best uniform approxi-mation of the function f , defined by the relation

ER(f) = minM

‖f −M‖C , where M(x) =∑

‖k‖6Rck e

ik·x.

Thus, we obtain the statement: if the function f continuous in Rm and 2π-periodic ineach variable is such that HR,pER(f) −→

R→+∞0, then

maxx

1

n

n−1∑

j=0

|Sj(f, x)− f(x)|p −→n→+∞

0 and maxx

1

R

∫ R

0|Sr(f, x)− f(x)|pdr −→

R→+∞0.

REFERENCES

[1] G. H. Hardy and J. E. Littlewood, Sur la série de Fourier d’une fonction àcarré sommable, C. R. Acad. Sci., Paris, 156 (1913), 1307–1312.

[2] O. I. Kuznetsova, Strong spherical averages and L-convergence of multipletrigonometric series, Dokl. Akad. Nauk, 391 (2003), 303–305.

[3] O. I. Kuznetsova, Strong spherical averages of multiple trigonometric series, Izv.Nat. Akad. Nauk Armenii, 44 (2009), 27–40.

[4] O. I. Kuznetsova and A. N. Podkorytov, On strong spherical averages ofFourier series, Algebra i Analiz, 25 (2013).

Institute for Applied Mathematics and Mechanics, NAS UkraineE-mail: [email protected]. Petersburg State UniversityE-mail: [email protected]

1We write αn≍βn if αn=O(βn) and βn=O(αn). The constants in the inequalities that are true forall n may depend only on the dimension m.

81


ON CONGRUENCE EXTENSION PROPERTYFOR ORDERED ALGEBRAS

Valdis Laan

Let Ω be a type. An ordered Ω-algebra is a triplet A = (A,ΩA,≤A) comprising aposet (A,≤A) and a set ΩA of operations on A such that all the operations ωA ∈ ΩA aremonotone mappings. An order-congruence on A is an algebraic congruence θ on A suchthat aθa′ whenever

a ≤ a1θa2 ≤ . . . ≤ an−1θan ≤ a′ ≤ a′1θa′2 ≤ . . . ≤ a′m−1θa

′m ≤ a

for some a1, . . . , an, a′1, . . . , a′m ∈ A. By a precongruence on A we mean a preorder on Awhich is compatible with operations and extends the order of A.

We say that an ordered algebra A has the congruence extension property if everyorder-congruence θ on an arbitrary subalgebra B of A is induced by an order-congruenceΘ on A, i.e. Θ ∩ (B ×B) = θ. We say that an ordered algebra A has the precongruenceextension property if every precongruence σ on an arbitrary subalgebra B of A is inducedby a precongruence Σ on A.

We discuss some results and open problems about these two properties.The research was partially supported by Estonian Science Foundation Grant 8394

and Estonian Targeted Financing Project SF0180039s08.


82


ON DUALITY OF DIAMETER 2 PROPERTIES

Rainis Haller, Johann Langemets, and Märt Põldvere

We study octahedrality and roughness of norms on Banach spaces and show thatthey are dual notions to diameter 2 properties (see [ALN]).

Octahedral norms were introduced by Godefroy and Maurey [GM] (see also [G]) inorder to characterize Banach spaces containing an isomorphic copy of ℓ1. The connectionof octahedral norms to the subject appears probably first in Deville’s paper [D] where itis proven that an (everywhere) octahedral norm is 2-average rough. We consider stabilityresults for octahedrality and derive some known results for diameter 2 properties moreconveniently.


REFERENCES

[ALN] T. Abrahamsen, V. Lima, and O. Nygaard, Remarks on diameter 2 proper-ties, J. Conv. Anal. 20 (2013), 439–452.

[D] R. Deville, A dual characterisation of the existence of small combinations ofslices, Bull. Austral. Math. Soc. 37 (1988), 113–120.

[G] G. Godefroy, Metric characterization of first Baire class linear forms and oc-tahedral norms, Studia Math. 95 (1989), 1–15.

[GM] G. Godefroy and B. Maurey, Normes lisses et normes anguleuses sur lesespaces de Banach séparables, unpublished preprint.

Faculty of Mathematics and Computer Science, University of Tartu, 50409 Tartu, J. Liivi2, EstoniaE-mails: [email protected], [email protected], [email protected]

83


THE POSITIVITY PROBLEMFOR LINEAR RECURRENCE SEQUENCES

Vichian Laohakosol

A (real) linear recurrence sequence (un)n≥0 of order k ∈ N, k ≥ 2, is a sequencesatisfying

un = a1un−1 + a2un−2 + · · ·+ akun−k (n ≥ k), (3)

where a1, a2, . . . , ak ( 6= 0) and the initial values u0, u1, . . . , uk−1 are given real numbers.Two important decision problems related to linear recurrence sequences are as follows.

• The Skolem Problem: does a given linear recurrence sequence have a zero?

• The Positivity Problem: are all the terms of a given linear recurrence sequencepositive?

It is known that the decidability of the Positivity Problem implies the decidability of theSkolem Problem. At present, the decidability of each of these problems remains open.However, certain partial results have already appeared, viz., the Positivity Problem forsequences satisfying a second order linear recurrence relation was shown to be decid-able by Halava–Harju–Hirvensalo, [1], in 2006. The Positivity Problem for sequencessatisfying a third or a fourth order linear recurrence relation was shown to be decidableby Laohakosol–Punnim–Tangsupphathawat, in [2], and [3] and [4], respectively. Ourobjective is to discuss the decidability of the Positivity Problem, especially of low orders.

REFERENCES

[1] V. Halava, T. Harju, and M. Hirvensalo, Positivity of second order linearrecurrent sequences, Discrete Appl. Math. 154 (2006), 447–451.

[2] V. Laohakosol and P. Tangsupphathawat, Positivity of third order linear re-currence sequences, Discrete Appl. Math. 157 (2009), 3239–3248.

[3] P. Tangsupphathawat, N. Punnim, and V. Laohakosol, The positivity problemfor fourth order linear recurrence sequences is decidable, Colloq. Math. 128 (2012),133–142.

[4] P. Tangsupphathawat, N. Punnim, and V. Laohakosol, Another proof ofthe positivity problem for fourth order recurrence sequences, East-West J. Math. 14(2012), 185–200.

Departmet of Mathematics, Kasetsart University, Bangkok 10900, ThailandE-mail: [email protected]

84


MULTIPLE FOURIER EXPANSIONS OVERWALSH–PALEY AND TRIGONOMETRIC SYSTEMS

Svetlana Bloshanskaya, Igor Bloshanskii, and Olga Lifantseva

We consider IN = [0, 1)N , N ≥ 2, and two orthonormal on IN systems Ψ = E and Ψ =W (where E = ei2πnxn∈ZN is the multiple trigonometric system, andW = wn(x)n∈ZN

0

is the multiple Walsh–Paley system, x ∈ IN , ZN0 = n ∈ ZN : nj ≥ 0, j = 1, . . . , N).Let E be an arbitrary measurable set, E ⊂ IN , µE > 0 (µ = µN is the N -dimensional

Lebesgue measure), and let A = A(IN ) be a linear subspace of L1(IN ).We investigate the behavior of rectangular partial sums Sn(x, f ; Ψ) of multiple Fourier

series over the system Ψ (here x ∈ IN , f ∈ A(IN ), n ∈ ZN0 ) as n → ∞, i.e., min1≤j≤N

nj →∞, on the sets E and IN \E, depending on the smoothness of the function f (i.e., on thetype of the space A), on structural and geometric characteristics of the set E (SGC(E)),as well as on restrictions imposed on the components n1, . . . , nN of the vector n (the“index” of the partial sum Sn(x, f ; Ψ)).

In [1] for a wide class of measurable sets E, E ⊂ IN , N ≥ 3, we have obtaineda criterion for validity (in terms of SGC(E)) of weak generalized localization almosteverywhere of the considered Fourier series (i.e., the necessary and sufficient conditions forthe convergence almost everywhere on some set E1, E1 ⊂ E, µE1 > 0, of the consideredseries, when the expanded function equals zero on E), for the case when A(IN ) = Lp(IN ),p > 1, and partial sums Sn(x, f ; Ψ) have “index” n = (n1, . . . , nN ), in which some of thecomponents are elements of (single) lacunary sequences (i.e. for some components nj ofthe vector n, the following conditions are satisfied:

n(s+1)j

n(s)j

≥ q > 1, s = 1, 2, . . . ).

The problem indicated above was investigated by us in the Orlicz spaces as well.We have also shown how the results formulated above can be “sewed” with the resultsobtained earlier in [2, 3].

This work is supported by Grant 11-01-00321 of Russian Foundation for Basic Re-search.

REFERENCES

[1] S. K. Bloshanskaya, I. L. Bloshanskii, and O. V. Lifantseva, Trigonomet-ric Fourier series and Walsh-Fourier series with lacunary sequence of partial sums,Matem. Zametki 93 (2013), 305–309; English transl. in Math. Notes 93 (2013),332–336.

85

[2] I. L. Bloshanskii, On criteria for weak generalized localization in N -dimensionalspace, Dokl. Akad. Nauk SSSR 271 (1983), 1294–1298; English transl. in SovietMath. Dokl. 28 (1983).

[3] S. K. Bloshanskaya and I. L. Bloshanskii, Weak generalized localization formultiple Walsh–Fourier series of functions in Lp, p ≥ 1, Tr. Mat. Inst. Steklova,214 (1997), 83–106.

Department of Higher Mathematics, National Research Nuclear University, 115409 Moscow,31 Kashirskoye Shosse, RussiaE-mail: [email protected] of Mathematical Analysis and Geometry, Moscow State Regional University,105005 Moscow, 10a Radio Street, RussiaE-mails: [email protected], [email protected]

86


FOURIER TRANSFORMVERSUS HILBERT TRANSFORM

Elijah Liflyand

We present several results in which the interplay between the Fourier transform andthe Hilbert transform is of special form and importance. Known relations between thetwo operators are supplied with new ones.

1. In the 50s (Kahane, Izumi–Tsuchikura, Boas, etc.; see [2]), the following problemin Fourier Analysis attracted much attention:

Let ak∞k=0 be the sequence of the Fourier coefficients of the absolutely convergentsine (cosine) Fourier series of a function f : T = [−π, π) → C, that is

∑ |ak| < ∞.Under which conditions on ak the re-expansion of f(t) (f(t) − f(0), respectively) inthe cosine (sine) Fourier series will also be absolutely convergent?

We solve a similar problem for functions on the whole axis and their Fourier trans-forms. Generally, the re-expansion of a function with integrable cosine (sine) Fouriertransform in the sine (cosine) Fourier transform is integrable if and only if not only theinitial Fourier transform is integrable but also the Hilbert transform of the initial Fouriertransform is integrable.

We observe that a similar answer is true for Fourier series in terms of the discreteHilbert transform, contrary to the known results that were just sufficient conditions forthe summability of the discrete Hilbert transform.

Comparing these two settings, one arrives to the necessity of obtaining effective suf-ficient conditions for the integrability of the Hilbert transform. Known conditions arediscussed and new ones are obtained.

2. The following result is due to Hardy and Littlewood (see, e.g., [4]).If a (periodic) function f and its conjugate f are both of bounded variation, their

Fourier series converge absolutely.We generalize the Hardy–Littlewood theorem (joint work with U. Stadtmüller) to

functions on the real axis, hence the absolute convergence of the Fourier series should bereplaced by the integrability of the Fourier transform.

Since a function f of bounded variation may be not integrable, its Hilbert transform,a usual substitute for the conjugate function, may not exist. One has to use the modifiedHilbert transform

f(x) = (P.V.)1

π

∫

Rf(t)

1

x− t+

t

1 + t2

dt.

Theorem. Let f be a function of bounded variation and vanish at infinity: lim|t|→∞

f(t) =

0. If its conjugate f is also of bounded variation, then the Fourier transforms of bothfunctions are integrable on R.

87

The initial Hardy–Littlewood theorem is a partial case of this extension, when thefunction is taken to be of compact support.

A corresponding extension to radial functions readily follows from the obtained one-dimensional results.

3. In various of the considered problems, one of the main tools is the well-knownextension of Hardy’s inequality:

∫

R

|g(x)||x| dx . ‖g‖H1(R).

It turns out that if the left-hand side is finite, then g is the derivative of a functionof bounded variation f which is locally absolutely continuous, lim

|t|→∞f(t) = 0, and its

Fourier transform is integrable.This is one of the strongest justifications for systematic investigation of the Fourier

transform of a function of bounded variation.We have found the maximal space for the integrability of such a Fourier transform,

it is inspired by work [1]. To be more precise, we let the (integrable) derivative of thegiven function belong to the class of integrable functions g for which the left-hand sidein Hardy’s inequality is integrable as well.

Along with those known earlier (see, e.g., [3]), various interesting new spaces ap-pear in this study. Their inter-relations lead, in particular, to improvements of Hardy’sinequality.

REFERENCES

[1] R. L. Johnson and C. R. Warner, The convolution algebra H1(R), J. FunctionSpaces Appl. 8 (2010), 167–179.

[2] J.-P. Kahane, Séries de Fourier absolument convergentes , Springer, Berlin, 1970.

[3] E. Liflyand, Fourier transforms of functions from certain classes , Anal. Math. 19(1993), 151–168.

[4] A. Zygmund, Trigonometric Series , Vols. I, II, Cambridge Univ. Press, Cambridge,1968.

Department of Mathematics, Bar-Ilan University, 5290002 Ramat-Gan, IsraelE-mail: [email protected]

88


SOME PROPERTIES OF GENERATING SYSTEMSOF SETS AND SEQUENCES

Rauni Lillemets

The notions of a generating system of sets and a generating system of sequences wereintroduced in 1980 by Irmtraud Stephani in [1].

We try to further understand the underlying structure and connections between thosetwo notions. We also introduce a way to construct a wide range of generating systems ofsets provided we are given a class of sequences that satisfy certain conditions.

The talk is based on author’s master thesis supervised by Eve Oja.The research was partially supported by Estonian Science Foundation Grant 8976.

REFERENCES

[1] I. Stephani, Generating systems of sets and quotients of surjective operator ideals,Math. Nachr. 99 (1980), 13–27.


89


ESSENTIAL NORM ESTIMATESFOR COMPOSITION OPERATORS ON BMOA

Mikael Lindström

In this talk I will discuss two function-theoretic estimates for the the essential normof an arbitrary composition operator Cϕ : f 7→ f ϕ acting on the space BMOA, whereϕ is an analytic self-map of the open unit disc in the complex plane: one in terms ofthe n-th power ϕn of the symbol ϕ and one which involves the Nevanlinna countingfunction. We also discuss new estimates in the special cases of symbols ϕ which belongto the subspace VMOA or which are univalent.

REFERENCES

[1] P. Galindo, J. Laitila, and M. Lindström, Essential norm estimates for com-position operators on BMOA, J. Funct. Anal., 265 (2013), 629–643.

Department of Mathematical Sciences, University of Oulu, FinlandE-mail: [email protected]

90


REMARKS ON ESSENTIALLY MINIMAL CLONES

Hajime Machida

For a non-empty set A, a clone C on A is essentially minimal if C is not a minimalclone but is minimal among all clones on A containing essential functions. For an es-sentially minimal clone C the rank of C is the least arity of the generators of C. For afinite set A, the rank of any essentially minimal clone on A is shown to be no greaterthan |A|. For a three-element set E3, all conjugate classes of essentially minimal cloneson E3 whose rank is 2 are determined. Some examples of essentially minimal clones onE3 whose rank is 3 will also be presented.

This is a joint work with Ivo G. Rosenberg (Montréal).

REFERENCES

[1] H. Machida and I. G. Rosenberg, Classifying essentially minimal clones, Pro-ceedings 14th ISMVL, IEEE (1984), 4–7.

[2] H. Machida and I. G. Rosenberg, Essentially minimal groupoids, in “Algebrasand Orders”, Kluwer Academic Publishers (1993), 287–316.

[3] H. Machida and I. G. Rosenberg, A study on essentially minimal clones, Pro-ceedings 43rd ISMVL, IEEE (2013), 117–122.

International Christian University, Mitaka, Tokyo, JapanE-mail: [email protected]

91


MEASURES OF NONCOMPACTNESSAND SOME APPLICATIONS

Eberhard Malkowsky

Measures of noncompactness have important applications in various parts of math-ematics, in particular, in functional analysis and operator theory. In this talk we givean introduction to measures of noncompactness on bounded sets of complete metricspaces and Banach spaces, and a survey of the most important basic properties of cer-tain measures of noncompactness, in particular, we consider the Kuratowski, Hausdorffand separation measures of noncompactness. We also define and study the Hausdorffmeasure of noncompactness of operators between Banach spaces. Finally, we deal withsome applications of measures of noncompactness.

Department of Mathematics, Faculty of Science, Fatih University, Büyükçekmece 34500,Istanbul, Turkey; Department of Mathematics, University of Giessen, Arndtstrasse 2,D–35392 Giessen, GermanyE-mails: [email protected], [email protected]

92


COMMUTATIVE ORDERS IN SEMIGROUPS

László Márki

We consider commutative orders, that is, commutative semigroups having a semi-group of quotients in a local sense defined as follows. An element a ∈ S is square-cancellable if for all x, y ∈ S1 we have that xa2 = ya2 implies xa = ya and also a2x = a2yimplies ax = ay. It is clear that being square-cancellable is a necessary condition for anelement to lie in a subgroup of an oversemigroup. In a commutative semigroup S, thesquare-cancellable elements constitute a subsemigroup S(S). Let S be a subsemigroupof a semigroup Q. Then S is a left order in Q and Q is a semigroup of left quotients of Sif every q ∈ Q can be written as q = a♯b where a ∈ S(S), b ∈ S and a♯ is the inverse ofa in a subgroup of Q and if, in addition, every square-cancellable element of S lies in asubgroup of Q. Right orders and semigroups of right quotients are defined dually. If S isboth a left order and a right order in Q, then S is an order in Q and Q is a semigroup ofquotients of S. We remark that if a commutative semigroup is a left order in Q, then Qis commutative so that S is an order in Q. A given commutative order S may have morethan one semigroup of quotients. The semigroups of quotients of S are pre-ordered bythe relation Q ≥ P if and only if there exists an onto homomorphism φ : Q → P whichrestricts to the identity on S. Such a φ is referred to as an S-homomorphism; the classesof the associated equivalence relation are the S-isomorphism classes of orders, giving usa partially ordered set Q(S). In the best case, Q(S) contains maximum and minimumelements. In a commutative order S, S(S) is also an order and has a maximum semigroupof quotients R, which is a Clifford semigroup. We investigate how much of the relationbetween S(S) and its semigroups of quotients can be lifted to S and its semigroups ofquotients.

This is a joint work with P. N. Ánh, V. Gould, and P. A. Grillet.

Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences, BudapestE-mail: [email protected]

93


DERIVATIONS AND LOCAL MULTIPLIERSOF C*-ALGEBRAS

Martin Mathieu

We discuss the interrelation between derivations and local multipliers of C*-algebrasand report on some recent advances on a problem posed by Gert Pedersen back in 1978.

Department of Pure Mathematics, Queen’s University Belfast, Belfast BT7 1NN, North-ern IrelandE-mail: [email protected]

94


HARDY INEQUALITIESAND INTERPOLATION OF LORENTZ SPACESASSOCIATED TO A VECTOR MEASURE (II)

Ricardo del Campo, Antonio Fernández, Antonio Manzano,Fernando Mayoral, and Francisco Naranjo

Let m be a vector measure, ρ a parameter function, 1 < p < ∞, and X an inter-mediate quasi-Banach space between the spaces Lp,1 and Lp,∞ associated to the semi-variation ‖m‖ of m. The aim of this talk is to analyze the role played by the Ariño–Muckenhoupt classes to determine conditions under which the real interplation space(X,L∞(m))ρ,q is equal to the Lorentz type space Λqϕp(‖m‖) associated to the weight ϕp

given by ϕp(t) := t1p

ϕ(t1p )

. Note that the case p = 1 is precisely what has been developed

in our first talk.This research is partially supported by M.E.C. (Spain) and FEDER under projects

MTM2009-14483, MTM2010-15814 and MTM2012-36740.

REFERENCES

[1] M. A. Ariño and B. Muckenhoupt, Maximal functions on classical Lorentzspaces and Hardy’s inequality with weights for nonincreasing functions, Trans. Amer.Math. Soc. 320 (1990), 727–735.

[2] J. Bergh and J. Löfström, Interpolation spaces, An introduction, Springer,Berlin, 1976.

[3] J. Gustavsson, A function parameter in connection with interpolation of Banachspaces, Math. Scand. 42 (1978), 289–305.

[4] L. E. Persson, Interpolation with a parameter function, Math. Scand. 59 (1986),199–222.

[5] E. Sawyer, Boundedness of classical operators on classical Lorentz spaces, StudiaMath. 96 (1990), 145–158.

Escuela Técnica Superior de Ingeniería Agronómica, Universidad de Sevilla, Carreterade Utrera, km 1, 41013, Sevilla, SpainE-mail: [email protected]

95

Escuela Técnica Superior de Ingenieros, Universidad de Sevilla, Camino de los des-cubrimientos, 41092, Sevilla, SpainE-mails: [email protected], [email protected], [email protected] Politécnica Superior, Universidad de Burgos, C/ Villadiego, 09001, Burgos,SpainE-mail: [email protected]

96


THE DEGREE OF OPERATIONS ON GROUPS

Peter Mayr

Institute for Algebra, Johannes Kepler University Linz, Altenberger Str. 69, 4040 Linz,AustriaE-mail: [email protected]

In 1979 Harold Ward introduced the combinatorial degree of operations on abeliangroups as some generalization of the degree of polynomials on commutative rings. Weextend his notion further to arbitrary groups. In particular, this allows us to characterizefinite p-groups as those groups on which all operations have finite degree. The proof ofthat result uses some basic facts of group representation theory.

As an application we obtain efficient algorithms for several computational problems(membership, size, . . . ) on subalgebras of direct powers of groups with additional oper-ations.

97


THE VARIATION DETRACTING PROPERTYOF SOME SHANNON SAMPLING SERIESAND THEIR DERIVATIVES

Tarmo Metsmägi

We consider the generalized Shannon sampling operators, which preserve the totalvariation of functions and their derivatives. The generalized Shannon sampling operatorsfor the uniformly continuous and bounded functions on the real line, f ∈ C(R), are givenby (t ∈ R;W > 0)

(SW f)(t) :=∞∑

k=−∞f(

k

W)s(Wt− k),

where the kernel s(t) := s(λ; t) :=1∫0

λ(u) cos(πtu) du, and λ ∈ C[−1,1] is an even window

function, λ(0) = 1, λ(u) = 0 (|u| > 1).Proposition. Define the related kernel to the kernel s as follows:

sm1,...,mn(t) :=

∫ 1

0

λ(u)

sinc(m1u) . . . sinc(mnu)cos(πtu)du (1)

for 0 < m1, . . . ,mn ≤ 1 and sinc(u) := sin(πu)πu .Then

s(t) =1

2nm1 . . .mn

∫ m1

−m1

dx1

∫ m2

−m2

dx2 . . .

∫ mn

−mn

sm1,...,mn(t+ x1 + . . .+ xn)dxn.

Let BV (R) denote the class of all functions of bounded variation on R. The corre-sponding total variations are denoted by VR[f ]. The variation detracting property forderivatives reads as follows.Theorem. Assume that the kernel (1) satisfies sm1,...,mn+1 ∈ L1(R) for 0 < m1, . . . ,mn+1 ≤1 such that there exists b ∈ R with ±m1 ± . . . ± mn+1 − b ∈ Z. If f is bounded andf (n) ∈ BV (R), then (SW f)

(n) ∈ BV (R) and

VR[(SW f)(n)] ≤ ‖sm1,...,mn+1‖1 VR[f (n)].

This research was partially supported by Estonian Science Foundation Grant 8627.

Institute of Mathematics and Natural Sciences, Tallinn University, 10120 Tallinn, 25Narva Road, EstoniaE-mail: [email protected]

98


A NOTE ON EXTREME POINTS OF C∞-SMOOTH BALLSIN POLYHEDRAL SPACES

Vicente Montesinos

Morris [Mo83] proved that every separable Banach space X that contains an isomor-phic copy of c0 has an equivalent strictly convex norm such that all points of its unitsphere SX are unpreserved extreme, i.e., they are no longer extreme points of BX∗∗ . Weuse a result of Hájek [Ha95] to prove that any separable infinite-dimensional polyhedralBanach space has an equivalent C∞-smooth and strictly convex norm with the sameproperty as in Morris’ result. We additionally show that no point on the sphere of aC2-smooth equivalent norm on a polyhedral infinite-dimensional space can be stronglyextreme, i.e., for any such x, there exists a sequence (hn) in X with ‖hn‖ 6→ 0 such that‖x± hn‖ → 1.

This is a joint work with A. J. Guirao (Valencia) and V. Zizler (Calgary).

REFERENCES

[GMZ] A. J. Guirao, V. Montesinos, and V. Zizler, A note on extreme pointsof C∞-smooth balls in polyhedral spaces, Preprint.

[Ha95] P. Hájek, Smooth norms that depend locally on a finite number of coordinates,Proc. Amer. Math. Soc. 123 (1995), 3817–3821.

[HMZ12] P. Hájek, V. Montesinos, and V. Zizler, Geometry and Gâteaux smooth-ness in separable Banach spaces, Oper. Matrices 6 (2012), 201–232.

[Mo83] P. Morris, Dissapearance of extreme points, Proc. Amer. Math. Soc. 88(1983), 244–246.

Universidad Politécnica de ValenciaE-mail: [email protected]

99


COMPLEX INTERPOLATION OF Lp-SPACESOF INTEGRABLE FUNCTIONSWITH RESPECT TO VECTOR MEASURES

Ricardo del Campo, Antonio Fernández, Fernando Mayoral,and Francisco Naranjo

It is well-known that if (Ω,Σ) is a measurable space, µ a σ-finite scalar measure,0 < θ < 1 ≤ p0 6= p1 ≤ ∞ then the interpolated space between the spaces Lp0(µ)and Lp1(µ) by the first and the second Calderón methods is the space Lp1(µ), where1p = 1−θ

p0+ θ

p1, that is

[Lp0(µ), Lp1(µ)][θ] = [Lp0(µ), Lp1(µ)][θ] = Lp(µ). (1)

In this talk we will answer the following question:What happens with the equalities (1) when we consider Lp-spaces of integrable func-

tions with respect to a vector measure?

REFERENCES

[1] A. Fernández, F Mayoral, F. Naranjo, C. Sáez, and E. A. Sánchez-Pérez, Spaces of p-integrable functions with respect to a vector measure, Positivity10 (2006), 1–16.

[2] R. del Campo, A. Fernández, F Mayoral, F. Naranjo, and E. A. Sánchez-Pérez, Complex interpolation of spaces of integrable function with respect to a vectormeasure, Collect. Math. 61 (2010), 241–252.

[3] R. del Campo, A. Fernández, F. Mayoral, and F. Naranjo, Complex in-terpolation of Lp-spaces of vector measures on δ-rings, J. Math. Anal. Appl. 405(2013), 518–529.

ETSIA, University of Seville, Ctra. de Utrera Km. 1, 41013–Sevilla, Spain.E-mail: [email protected], University of Seville, Camino de los Descubrimientos, s/n, 41092-Sevilla, Spain.E-mails: [email protected], [email protected], [email protected]

100


LIPSCHITZ FUNCTIONS AND M-IDEALS

Heiki Niglas, Eve Oja, and Indrek Zolk

In [BW], Heiko Berninger and Dirk Werner tried to answer the following question.Berninger–Werner problem. Is the little Hölder space lip([0, 1]α) an M -ideal in theHölder space Lip([0, 1]α) for every α ∈ (0, 1)?

They showed that lip([0, 1]α) is anM -ideal in a non-separable subspace of Lip([0, 1]α),whilst they conjectured that the answer to the problem might be negative.

In [K, Theorem 6.6], Nigel J. Kalton proved that for a compact metric space M andevery ǫ > 0, the little Lipschitz space lip(M) is (1+ǫ)-isomorphic to a subspace of c0.

As it is well known that lip(Mα) is a canonical predual of F(Mα) for a compactmetric space M , Kalton solved the Berninger–Werner problem in full generality, whichmeans that the following holds: if M is a compact metric space, then the little Hölderspace lip(Mα) is an M -ideal in the Hölder space Lip(Mα).

We show how to use Kalton’s theorem to prove some results concerning properties ofthe spaces lip(M), K(lip(M)), and L(lip(M)) for a compact metric space M . We alsopresent some further applications.


REFERENCES

[BW] H. Berninger and D. Werner, Lipschitz spaces and M-ideals, Extracta Math.18 (2003), 33–56.

[K] N. J. Kalton, Spaces of Lipschitz and Hölder functions and their applications,Collect. Math. 55, 2 (2004), 171–217.

Faculty of Mathematics and Computer Science, University of Tartu, 50409 Tartu, J. Liivi2, EstoniaE-mail: [email protected] of Mathematics and Computer Science, University of Tartu, 50409 Tartu, J. Liivi2, Estonia; Estonian Academy of Sciences, 10130 Tallinn, Kohtu 6, EstoniaE-mail: [email protected] of Mathematics and Computer Science, University of Tartu, 50409 Tartu, J. Liivi2, EstoniaE-mail: [email protected]

101


LYAPUNOV THEOREMFOR q-CONCAVE BANACH SPACES

Anna Novikova

Let X be a Banach space, (Ω,Σ) be a measure space, where Ω is a set and Σ isa σ-algebra of subsets of Ω. If m : Σ → X is a σ-additive X-valued measure, then therange of m is the set m(Σ) := m(A) : A ∈ Σ. The measure m is non-atomic if, forevery set A ∈ Σ with m(A) > 0, there exist B ⊂ A,B ∈ Σ, such that m(B) 6= 0 andm(A\B) 6= 0. We will call an X-valued measure a Lyapunov measure if the closure of itsrange is convex. The space X is a Lyapunov space if every X-valued non-atomic measureis Lyapunov. The spaces ℓp, 1 ≤ p <∞, p 6= 2, and c0 are Lyapunov spaces [1].

The following result is a generalization of the famous Lyapunov theorem for Rn-valuedmeasures [2].

Theorem. Let X be a q-concave (for some q <∞) Banach space with unconditionalbasis which does not contain an isomorphic copy of ℓ2. Then X is a Lyapunov space.

The proof uses some results from [3].

REFERENCES

[1] V. Kadets and G. Schechtman, Lyapunov’s theorem for lp-valued measures, St.Petersburg Math. J. 4 (1993), 916–965.

[2] A. Lyapunov, Sur les fonctions-vecteurs complètement additives, Izv. Akad. NaukSSSR Ser. Mat. 4 (1940), 465–478.

[3] V. Mykhaylyuk, M. Popov, B. Randianantoanina, and G. Schechtman,Narrow and l2-strictly singular operators from Lp, submitted.

Faculty of Mathematics and Computer Sciences, Weizmann Institute of Sciences, 76100Rehovot, POB 26, IsraelE-mail: [email protected]

102


THE SYMMETRIC STRONG DIAMETER 2 PROPERTYIN BANACH SPACES

Olav Nygaard

The diameter 2 property (D2P) for a Banach space X means that every non-voidrelatively weakly open set of the unit ball BX has diameter 2. In particular, every slice(of BX) then has diameter 2. A property that implies the D2P (and much more) isthe following: X is said to enjoy the symmetric strong diameter 2 property if whenever(Si(x

∗i , εi))

ni=1 are n slices of BX and ε > 0, there exist xi ∈ Si and ϕ ∈ BX such that

xi ± ϕ ∈ Si, i = 1, 2, . . . , n, and ‖ϕ‖ > 1 − ε. Let us call this property the symmetricstrong diameter 2 property (SSD2P) – a motivation for that particular name will be givenduring the talk.

We will give examples of classes of Banach spaces having the SSD2P, discuss (lack of)stability when forming p-sums, look at possible passage of SSD2P to subspaces, see thatLindenstrauss spaces have the SSD2P, and finally, ask some natural questions arising.

The talk is based on a preliminary joint work and discussions with Rainis Haller,Johann Langemets and Märt Põldvere, University of Tartu.

Department of Mathematics, University of Agder, Servicebox 422, 4604 Kristiansand,NorwayE-mail: [email protected]

103


ON APPROXIMATION PROPERTIESOF KANTOROVICH-TYPE SAMPLING OPERATORS

Olga Orlova and Gert Tamberg

For the uniformly continuous and bounded functions f ∈ C(R) the generalized sam-pling operators SW and the corresponding Kantorovich-type sampling operators SKW,n(cf. [1]) (n ∈ N) are given by (t ∈ R; W > 0)

(SW f)(t) :=∞∑

k=−∞f(

k

W)s(Wt− k),

(SKW,nf)(t) :=

∞∑

k=−∞

nW

(2nk+1)/2nW∫

(2nk−1)/2nW

f(u) du

s(Wt− k).

Since in many applications the results of measurements are some local averages, not theexact point estimates, it is more natural to consider Kantorovich-type sampling operatorsSKW,n instead of the operators SW .

We show that we can use the results we have for operators SW to prove analogousresults for operators SKW,n.

Theorem 1. If the sampling operator SW : C(R) → C(R) has the finite norm, i.e.

‖SW ‖ = supu∈R

∞∑

k=−∞|s(u− k)| <∞,

then the corresponding Kantorovich-type operator SKW,n has the norm ‖SKW,n‖ = ‖SW ‖(n ∈ N).Theorem 2. If we can estimate the order of approximation by the operator SW via themodulus of smoothness of order r > 2, then we have for the corresponding Kantorovich-type operator SKW,n the estimate

‖SKW,nf − f‖C 6Mω2(f, 1/W ).

This research was supported by Estonian Science Foundation Grant 9383 and Esto-nian Targeted Financing Project SF0140011s09.

104

REFERENCES

[1] C. Bardaro, G. Vinti, P. L. Butzer, and R. L. Stens, Kantorovich-type gen-eralized sampling series in the setting of Orlicz spaces, Sampl. Theory Signal ImageProcess. 6 (2007), 29–52.

Deptartment of Mathematics, Tallinn University of Technology, 19086 Tallinn, 5 Ehita-jate tee, EstoniaE-mails: [email protected], [email protected]

105


PRODUCTIVE PROPERTY IN TOPOLOGICAL INVERSESEMIGROUPS

Kateryna Pavlyk

A topological space X is said to be pseudocompact if each locally finite family ofnonempty open susbets of the space X is finite. The pseudocompactness is the propertyof topological space which is not even finitely multiplicative, but therewith it is the onlyproperty of topological groups preserved under multiplication of any number of factorshaving the corresponding property, the so-called Comfort-Ross theorem [1]. Ravskygeneralized this result for paratopological groups: the product of a nonempty family ofpseudocompact paratopological groups is pseudocompact [2].

A topological inverse semigroup is an inverse topological semigroup with continuousinversion. A topological inverse semigroup S is called primitive if S contains at leasttwo distinct idempotents and for any distinct idempotents e, f of S the product ef isequal to the smallest idempotent of S. We extend the result of Comfort and Ross to theclass of pseudocompact primitive topological inverse semigroups. Also we prove that theStone-Čech compactification of a pseudocompact primitive topological inverse semigroupis a compact primitive one.

This research was carried out with the support of the ESF and co-funded by MarieCurie Actions, grant ERMOS36.

REFERENCES

[1] W. W. Comfort and K. A. Ross, Pseudocompactness and uniform continuity intopological groups, Pacif. J. Math. 16:3 (1966), 483—496.

[2] A. Ravsky, Pseudocompact paratopological groups that are topological, preprint.

Institute of Mathematics, University of Tartu, Tartu, EstoniaE-mail: [email protected]

106


ON THE BOHNENBLUST–HILLE INEQUALITY

Daniel Pellegrino

The multilinear Bohnenblust–Hille inequality asserts that for each positive integer mthere is a constant Cm ≥ 1 such that

N∑

i1,...,im=1

∣∣T (ei1 , . . . , eim)∣∣ 2mm+1

m+12m

≤ Cm ‖T‖ ,

for all positive integers N and all m-linear forms T defined on ℓN∞ × · · · × ℓN∞. Thisinequality was rediscovered recently and now it is known that the precise informationon the growth of its constants plays an important role in different fields of Mathematicsand Physics. We will present recent results on the estimates for the constants Cm forreal and complex scalars. We show that, in contrast with the predictions from the last80 years, these constants have, at least, a subpolynomial growth.

This research was partially supported by CNPq and CAPES.

REFERENCES

[1] D. Diniz, G. A. Muñoz-Fernández, D. Pellegrino, and J. B. Seoane-Sepúlveda, The asymptotic growth of the constants in the Bohnenblust–Hille in-equality is optimal, J. Funct. Anal. 263 (2012), 415–428.

[2] D. Nuñez-Alarcón, D. Pellegrino, and J. B. Seoane-Sepúlveda, On theBohnenblust-Hille inequality and a variant of Littlewood’s 4/3 inequality, J. Funct.Anal. 264 (2013), 326–336.

[3] D. Nuñez-Alarcón, D. Pellegrino, J. B. Seoane-Sepúlveda, andD. M. Serrano-Rodríguez, There exist multilinear Bohnenblust–Hille constants(Cn)

∞n=1 with limn→∞(Cn+1 − Cn) = 0, J. Funct. Anal. 264 (2013), 429–463.

[4] D. Pellegrino and J. B. Seoane-Sepúlveda, New upper bounds for the con-stants in the Bohnenblust–Hille inequality, J. Math. Anal. Appl. 386 (2012), 300–307.

Depto de Matemática, Universidade Federal da Paraíba, 58.051-900, João Pessoa, Paraíba,BrazilE-mails: [email protected], [email protected]

107


BORNOLOGICAL ALGEBRASWHICH INDUCE A TOPOLOGY THAT GIVESA TOPOLOGICAL ALGEBRA STRUCTURE

Mati Abel and Reyna María Pérez-Tiscareño

A bornology on a set X is a collection B of subsets of X which satisfies the followingconditions:

(a) X =⋃B∈B

B;

(b) If B ∈ B and C ⊆ B, then C ∈ B;

(c) If B1, B2 ∈ B, then B1 ∪B2 ∈ B.

A bornology on a vector space over K where K is the field of real or complex numbersis called a vector bornology if the following conditions are satisfied:

(d) If B1, B2 ∈ B, then B1 +B2 ∈ B;

(e) If B ∈ B and λ ∈ K, then λB ∈ B;

(f)⋃

|λ|61

λB ∈ B for every B ∈ B.

Moreover, when X is an algebra we shall say that (X,B) is a bornological algebra if(X,B) satisfies the conditions (a)− (f) and

(g) If a ∈ X and B ∈ B, then aB,Ba ∈ B.

I will talk about some results which answer the following question.When does a bornological algebra (E,B) induce a topology τ on E such that (E, τ)

is a topological algebra?Moreover, I will give some results about bornological algebras which are inductive

limits of bornological algebras.

Institute of Mathematics, University of Tartu, 2 J. Liivi Str., 50409 Tartu, EstoniaE-mails: [email protected], [email protected]

108


TRACES OF OPERATORS AND THEIR HISTORY

Dedicated to the memory of Erhard Schmidt,born on January 1, 1876, in Tartu

Albrecht Pietsch

As every mathematician knows, the trace of a square matrix is defined to be the sumof all entries of the main diagonal. Extending this concept to the infinite-dimensionalsetting does not always work, since non-converging infinite series may occur. So one hadto identify those operators that possess something like a trace. In a first step, this wasdone for operators on the separable Hilbert space. The situation in Banach spaces turnedout to be much more complicated, as the missing approximation property causes a lot oftrouble. I will present an axiomatic approach in which operator ideals play a dominantrole. My considerations include also singular traces that – by definition – vanish on allfinite rank operators. Thanks to the discoveries of Connes, those traces became a usefultool in non-commutative geometry, in the theory of pseudo-differential operators, and inquantum mechanics. The lecture is intended to show that people who prefer to live inHilbert spaces heavily need Banach spaces techniques.

REFERENCES

[1] S. Lord, F. Sukochev, and D. Zanin, Singular Traces, De Gruyter, Berlin, 2012.

[2] A. Pietsch, History of Banach Spaces and Linear Operators, Birkhäuser, Boston,2007.

[3] A. Pietsch, Traces on operator ideals and related linear forms on sequence ideals(part I ), Indag. Math., to appear, DOI: 10.1016/j.indag.2012.08.008.

FSU Jena, Ernst-Abbe-Platz 2, 07743 Jena, GermanyE-mail: [email protected]

109


NECKLACES AND q-CYCLES

Umarin Pintoptang

Let n ≥ 2 be a positive integer and q be a prime power. Consider necklaces consistingof n beads each of which has one of the given q colors. A primitive Cn-orbit is anequivalence class of n necklaces closed under rotation. A primitive Cn-orbit is self-complementary when it is closed under color matching. In [4], it is shown that the 1− 1correspondence between the set of self-complementary primitive Cn-orbits and the set ofself-reciprocal irreducible monic (srim) polynomials.

Let N be positive integer with gcd(q,N) = 1. A q-cycle(N) is a finite sequence ofnon-negative integers closed under multiplication by q. In [5], it is shown that q-cycles(N)are closely related to monic irreducible divisors of xN − 1 in Fq[x].

Here we discuss the following:

(i) q-cycles(N) can be used to obtain information about srim-polynomials;

(ii) connection between q-cycles(N) and Cn-orbits;

(iii) alternative proof of Miller’s results mentioned above.

REFERENCES

[1] S. D. Cohen, Polynomials over finite fields with large order and level, Bull. KoreanMath. Soc. 24(2) (1987), 83–96.

[2] H. Meyn, On the construction of irreducible self-reciprocal polynomials over finitefields, Appl. Algebra Engrg. Comm. Comput (AAECC) 1 (1990), 43–53.

[3] R. Lidl and H. Niederreiter, Finite Fields, Cambridge Univ. Press, Cambridge,1997.

[4] R. L. Miller, Necklaces, symmetrices and self-reciprocal polynomials, DiscreteMathematics 22 (1978), 25–33.

[5] Z. X. Wan, Lectures on Finite Fields and Galois Rings, World Scientific, Singapore,2003.

[6] J. L. Yucas and G. L. Mullen, Self-reciprocal irreducible polynomials over finitefields, Des. Codes Cryptogr. 33 (2004), 275–281.

Department of Mathematics, Faculty of Science, Naresuan University, Phitsanulok 65000;Centre of Excellence in Mathematics, CHE, Bangkok 10400, ThailandE-mail: [email protected]

110


TWO APPROXIMATION PROPERTIES

Anatolij Plichko

We consider the following approximation properties.The first property takes origin in Numerical Mathematics, see e.g. [1]. A Banach space

X has norm approximation property if there is λ ≥ 1 such that for every ε > 0 and everyfinite-dimensional subspace E ⊂ X there is a finite dimensional operator T : X → Xwith ‖T‖ ≤ λ and such that for all x ∈ E

(1− ε)‖x‖ ≤ ‖Tx‖ ≤ (1 + ε)‖x‖.

A Banach space X has bounded separable approximation property (see e.g. [2]) if thereexists λ ≥ 1 such that for every finite set F ⊂ X there is a separable rank operatorT : X → X with ‖T‖ ≤ λ and Tx = x for each x ∈ F .

We show that the well-known Pisier space has not the norm approximation property.We know no Banach space without bounded separable approximation property.

REFERENCES

[1] G. Vainikko, Funktionalanalysis der Diskretisierungsmethoden, Teubner-Texte zurMathematik, B.G. Teubner Verlag, Leipzig, 1976.

[2] S. Berrios and G. Botelho, Approximation properties determined by operatorideals and approximability of homogeneous polynomials and holomorphic functions,Studia Math. 208 (2012) 97–116.

Department of Mathematics, Cracow University of Technology, 31-155 Cracow, 24 Warsza-wska Street, PolandE-mail: [email protected]

111


CLOSED RANGE COMPOSITION OPERATORSFOR ONE-DIMENSIONAL SMOOTH SYMBOLS

Adam Przestacki

Composition operators are one of the most natural operators acting on the space ofsmooth functions, which is very important for classical analysis. Deep understandingof their behaviour provides not only interesting information about themselves, but alsoabout the space of smooth functions and its elements.

The aim of the talk is to discuss the following problem: for which smooth symbols ψthe composition operator Cψ : C∞(R) → C∞(R), F 7→ F ψ, has closed range, i.e., whenthe set ImCψ = F ψ : F ∈ C∞(R) is closed in the space of smooth functions endowedwith the standard topology of uniform convergence of functions and all derivatives oncompact sets. We give a full characterization of such symbols. In particular we provethe following result.Theorem. Let ψ : R → R be a smooth function such that there are no points at whichall derivatives of ψ vanish. Then the composition operator Cψ : C∞(R) → C∞(R), F 7→F ψ, has closed range.

REFERENCES

[1] A. Przestacki, Composition operators with closed range for one-dimensionalsmooth symbols, J. Math. Anal. Appl. 399 (2013), 225–228.

[2] A. Przestacki, Characterization of composition operators with closed range forone-dimensional smooth symbols, Preprint.

Faculty of Mathematics and Computer Science, Adam Mickiewicz University in Poznań,Umultowska 87, 61-614 Poznań, PolandE-mail: [email protected]

112


EXTENSIONS OF SCHUR’S INEQUALITYFOR THE LEADING COEFFICIENTOF BOUNDED POLYNOMIALSWITH ONE PRESCRIBED ZERO

Heinz-Joachim Rack

Our point of departure is Schur’s Chebyshev-type inequality

|an| ≤ (cosπ

4n)2n2n−1, (1)

cf. [3, (16.4.6)] and [5, Theorem III*], for the leading coefficient of Pn ∈ Bn,0 = Pn :||Pn||∞,I ≤ 1 and Pn(−1) = 0, where Pn(x) =

∑nk=0 akx

k and || · ||∞,I denotes theuniform norm on I = [−1, 1]. Let Cn,0 denote the larger set of Pn’s, n ≥ 2, which arebounded by 1 only at the n+ 1 extremal points of Tn (Chebyshev polynomial on I) andfurthermore satisfy the asymmetric boundary condition Pn(−1) = 0 (the case Pn(1) = 0runs analogously). Our results include the following.

(I) Sharp V. A. Markov-type inequalities, cf. [3, Theorems 16.3.1; 16.3.2], for allcoefficients of Pn ∈ Cn,0, and their extremal polynomials (i.e., those for which theequality is attained) within Cn,0. In particular, the leading coefficient of Pn obeys

|an| ≤ (1− 1

2n)2n−1. (2)

This is in contrast to the unconstrained case (i.e., no prescribed zero on I), where both(1) and (2) would coincide with Chebyshev’s inequality |an| ≤ 2n−1, cf. [3, (16.3.2);(16.3.4)].

(II) Sharp Szegö-type inequalities for all consecutive pairs |ak−1|+ |ak| of coefficientsof Pn ∈ Cn,0, with n−k even, and their extremal polynomials within Cn,0. This result iscurious because the sharp upper bounds turn out to be the moduli of the correspondingcoefficients of Tn (as in Szegö’s inequality [3, Theorem 16.3.3]), although Tn is not amember of Cn,0, since Tn(−1) 6= 0. In particular, the leading pair obeys

|an−1|+ |an| ≤ 2n−1.

(III) Sharp Schur-type inequalities for all ak, with n − k even, if Pn ∈ Bn,0. Inparticular,

|an−2| ≤ (1− 2(n− 1)(sinπ

4n)4)(cos

π

4n)2(n−2)n2(n−2)−1.

The corresponding estimate for an−2 (according to (I)), if Pn ∈ Cn,0, is less involved:

|an−2| ≤ (1− n− 2

2n2)n2(n−2)−1.

113

A generalization to Pn(−1) = γ 6= 0 is possible. For alternative extensions of (1) see[1], [4]. Extensions of Schur’s second Chebyshev-type coefficient inequality [5, TheoremIV*] for polynomials which vanish symmetrically at both endpoints of I are given in [2].

REFERENCES

[1] M. A. Qazi and Q. I. Rahman, On a polynomial inequality of P. L. Chebyshev,Arch. Inequal. Appl. 1 (2003), 31–41.

[2] H.-J. Rack, Extensions of Schur’s inequality for the leading coefficient of boundedpolynomials with two prescribed zeros, pp. 107–116. In: Advances in Applied Math-ematics and Approximation Theory (G.A. Anastassiou and O. Duman, Eds.),Springer Proceedings in Mathematics & Statistics 41, Springer, New York, 2013.

[3] Q. I. Rahman and G. Schmeisser, Analytic Theory of Polynomials, London Math-ematical Society Monographs N. S. 26, Clarendon Press, Oxford, 2002.

[4] Q. I. Rahman and G. Schmeisser, Inequalities for polynomials on the unit inter-val, Trans. Am. Math. Soc. 231 (1977), 93–100.

[5] I. Schur, Über das Maximum des absoluten Betrages eines Polynoms in einemgegebenen Intervall, Math. Zeitschr. 4 (1919), 271–287.

Steubenstrasse 26a, 58097 Hagen, GermanyE-mail: [email protected]

114


EIGENVALUES OF (r, p)-NUCLEAR OPERATORSAND APPROXIMATION PROPERTIES OF ORDER (r, p)

Oleg I. Reinov

This lecture may be considered as a small survey of results obtained very recently byme and my PhD student Qaisar Latif. We consider different types of nuclear operatorsand investigate the corresponding eigenvalues problems, applying the results to obtaintrace-formulas of Grothendieck–Lidskiı type. For more or less complete investigationof the problems, we had to introduce some new notions of approximation propertiesfor Banach spaces providing both positive and negative results in connection with theproperties.

For example, we study tensor products and corresponding operator ideals of theoperators of type

(r; q) T =∑∞

n=1 x′n ⊗ yn, with one of the sequence being weakly q-summable while

another one is absolutely r-summable, for r ∈ (0, 1] (so, T is nuclear).In particular, for these cases, we have the following.(i) If q′ = p ∈ [1, 2] and T : X → X is as above, then for s > 0 with 1/s =

1/r + 1/2 − 1/p, T is of spectral type ls. For s = 1, the nuclear trace and the spectraltrace are well defined and coincide.

(ii) The result in (i) is sharp in a sense (roughly speaking, some approximation con-ditions are necessary). Discussing this sharpness, we give examples of operators U inBanach spaces, which are not nuclear, but whose adjoints U∗ are of type described in(r; q) (even, for 2/3 < r < 1).

The research was partially supported by the Higher Education Commission of Pak-istan and by grant 12-01-00216 of RFBR.

Faculty of Mathematics and Mechanics, St. Petersburg State University, 198504 St.Petersburg, Petrodvorets, 28 Universitetskii pr., RussiaE-mail: [email protected]

115


ON M(a,B, c)-IDEALS IN BANACH SPACES

Ksenia Rozhinskaya and Indrek Zolk

Let a, c ≥ 0 and let B be a compact set of scalars. A closed subspace Y of a Banachspace X is called an M(a,B, c)-ideal in X if there is a norm one projection P on X∗

such that kerP = Y ⊥ and

‖ax∗ + bPx∗‖+ c‖Px∗‖ ≤ ‖x∗‖ ∀b ∈ B, ∀x∗ ∈ X∗.

This approach was first suggested by E. Oja and it allows us to handle well-known specialcases of ideals, namely M -, h-, u- and M(r, s)-ideals (for definitions and references, see,e.g., [2]), in a more unified way.

We have developed easily verifiable equivalent conditions for a subspace of ℓ2∞ to bean M(a,B, c)-ideal.

Following what was done in [1] for M(r, s)-ideals, we obtain new results in a moregeneral M(a,B, c)-setting. Our main results are as follows. Suppose X and Y are closedsubspaces of a Banach space Z such that X ⊂ Y ⊂ Z. If X is an M(a,B, c)-ideal inY and Y is an M(d,E, f)-ideal in Z, then X is an ideal satisfying a certain type ofinequality in Z. Relying on this result, we show that if X is an M(a,B, c)-ideal in itssecond bidual, then X is an ideal satisfying a certain type of inequality in X(2n) for everyn ∈ N.

For illustration, we list here two corollaries of our results.

• If X is an M(a,B, c)-ideal in Y and Y is an M -ideal in Z, then X is an M(a,B, c)-ideal in Z.

• If X is a u-ideal in X∗∗, then X is an M(

12n−1 ,

− 2

2n−1

, 0)-ideal in X(2n) for

every n ∈ N.


REFERENCES

[1] R. Haller, On transitivity of M(r, s)-inequalities and geometry of higher duals ofBanach spaces, Acta Comment. Univ. Tartu. Math. 6 (2002), 9–13.

[2] E. Oja, Geometry of Banach spaces having shrinking approximations of the identity,Trans. Amer. Math. Soc. 352 (2000), 2801–2823.

Faculty of Mathematics and Computer Science, University of Tartu, J. Liivi 2, 50409Tartu, EstoniaE-mails: [email protected], [email protected]

116


TENSOR PRODUCTS OF Lp-SPACES ANDAPPLICATIONS TO DOMINATED POLYNOMIALSAND SPACES OF POLYNOMIALS

Pilar Rueda

The aim of the talk is to give a description of symmetric tensor products of (subspacesof) Lp-spaces as linear subspaces of Lq-spaces endowed with a suitable norm. As anapplication, we provide a factorization theorem for dominated polynomials through acanonical prototype of a dominated polynomial with values in a linear subspace of an Lp-space, endowed with a suitable norm. We also isolate the class of dominated polynomialswhich factor through a closed subspace of an Lp-space in the spirit of Pietsch.

The talk is based on some joint works with G. Botelho and D. Pellegrino, and withE. A. Sánchez-Pérez.

Dpto. Análisis Matemático, Universidad de Valencia, C/ Dr. Moliner 50, 46100 Bur-jassot (Valencia), SpainE-mail: [email protected]

117


APPROXIMATION BY TRIGONOMETRICBLACKMAN- AND ROGOSINSKI-TYPE OPERATORS

Anna Saksa and Andi Kivinukk

We study, for the 2π-periodic continuous functions f ∈ C2π, the trigonometric oper-ators

Un(f, x) =a02

+n∑

k=1

ϕ(k

n+ 1)(ak cos kx+ bk sin kx),

where ϕ ∈ C[0,1], ϕ(0) = 1, ϕ(1) = 0.Two types of operators will be defined:1. ϕa(t) = a+ 1

2 cosπt+ (12 − a) cos 2πt, a ∈ R, defines the Blackman-type operatorsBn,a : C2π → C2π;

2. ϕa(t) = a cos πt2 + (1 − a) cos 3πt2 , a ∈ R, defines the Rogosinski-type operators

Rn,a : C2π → C2π.Theorem 1. 1. If f ∈ C2π, then for any a ∈ R

‖Bn,af − f‖C2π ≤ (‖Bn,a‖+ |a|+ 1

2+ |1

2− a|)En(f) +

+1

4ω2(f,

π

n+ 1) +

|1− 2a|4

ω2(f,2π

n+ 1),

where En(f) is the best approximation, and ω2(f, δ) is the 2-nd modulus of continuity.2. In case a = 5/8, for any f ∈ C2π we have

‖Bn,5/8f − f‖C2π ≤ (5

4+ ‖Bn,5/8‖)En(f) +

1

16ω4(f,

π

n+ 1).

In some cases we are able to compute exact values of the operator norms ‖Bn,a‖ =sup‖Bn,af‖C2π : ‖f‖C2π ≤ 1.Theorem 2. 1. If 0 ≤ a ≤ 3/8, then

supn

‖Bn,a‖ = (1− 2a)(Sci(1) + Sci(4)) + 2a(Sci(2) + Sci(3)) =

= 1.064 . . .− a(0.159 . . .),

where Sci(x) = Si(πx)/π, Si() – the integral sine.2. In case a = 5/8, we have supn ‖Bn,5/8‖ = 1.234 . . .

Analogous results are valid for the Rogosinski-type operators.

This research was partially supported by the Estonian Science Foundation Grant8627.

Faculty of Mathematics, Tallinn University, 10120 Tallinn, 25 Narva road, EstoniaE-mails: [email protected], [email protected]

118


FACTORIZATION OF HOMOGENEOUS MAPSBETWEEN BANACH FUNCTION SPACESAND APPLICATIONS

Enrique A. Sánchez-Pérez

In this talk we present a new procedure for approximating linear operators betweenBanach function spaces. Our idea is to construct a framework for writing linear and con-tinuous maps as norm limits of bounded homogeneous maps which are characterized bymeans of a factorization diagram – equivalently, by a norm domination inequality. Usingthis, we show as an example a suitable approximation procedure for compact operatorsfrom Banach function spaces into Banach spaces as norm limits of homogeneous mapsthat satisfy a fixed factorization property. In fact, we prove that under the assumption ofcompactness of the linear map, it can be approximated by a sequence of order boundedand norm uniformly bounded homogeneous maps (no approximation properties of theinvolved spaces are needed). This setting is used for describing properties of the optimaldomain compact operators, i.e., the biggest Banach function space to which they can beextended. Our ideas have their roots in the classical Grothendieck description of weaklycompact sets in Banach spaces, but particular Banach lattice tools – p-convexification,Maurey–Rosenthal type theorems – are also used.

This is a joint research with Pilar Rueda (U. Valencia, Spain).

Instituto Universitario de Matemática Pura y Aplicada, Universidad Politécnica de Va-lencia, Spain.E-mail: [email protected]

119


Iλ-STATISTICALLY CONVERGENT SEQUENCESIN TOPOLOGICAL GROUPS

Ekrem Savas

Let λ = (λn) be a non-decreasing sequence of positive numbers tending to ∞ suchthat

λn+1 ≤ λn + 1, λ1 = 1.

In [1], P. Kostyrko et al. introduced the concept of I-convergence of sequences in ametric space and studied some properties of such convergence. Note that I-convergenceis an interesting generalization of statistical convergence.

We study Iλ-statistical convergence of sequences in topological groups and give someimportant inclusion theorems.

REFERENCES

[1] P. Kostyrko, T. Šalát, and W. Wilczynki, I-convergence, Real Anal. Exchange,26 (2000/2001), 669–685.

Istanbul Commerce University, Üküdar-Istanbul/TurkeyE-mail: [email protected]

120


ON STRONG AND STATISTICAL CONVERGENCESIN SOME FAMILIES OF SUMMABILITY METHODS

Anna Šeletski and Anne Tali

In our talk we discuss strong and statistical convergences in Cesàro- and Euler–Knopp-type families of summability methods.

Let x = (xn) be sequences with xn ∈ IC (n = 0, 1, 2, . . .) and let A be a transformationwhich transforms a sequence x into the sequence y = (yn). If the limit limn yn = s exists,then we say that x is convergent with respect to the summability method A (in short,A-convergent) to s. If yn = O(1), we say that x is bounded with respect to the methodA (in short, A-bounded).

Let Aα be a Cesàro- or an Euler–Knopp-type family of summability methods Aα(see [4]), where α is a continuous parameter with values α > α0 (α0 is some fixed realnumber). Suppose that [Aα+1]t are strong summability methods with positive exponentst = (tn) in the family Aα (see [3]).

We continue comparison of different strong summability methods [Aα+1]t in the familyAα, started in [3], with the help of a new convexity theorem. This convexity theoremgives sufficient conditions for the statement:

if x = (xn) is [Aβ+1]t-convergent to s and [Aγ+1]t-bounded, then x = (xn) is [Aδ+1]t-convergent to s for any β > δ > γ > α0 in a Cesàro-type family Aα.

The theorem mentioned above can be seen as a generalization of some convexitytheorems known earlier (see [5] for references).

Convergence of a sequence x = (xn) with respect to the different strong summabilitymethods [Aα+1]t in the family Aα is characterized also with the help of statisticalconvergence. Basing on papers [1] and [2] we compare Aα- and [Aα+1]t-convergences(and -boundednesses) of x with its A-statistical Aα-convergence (and, in particular, withits statistical Aα-convergence) for different values of parameter α, where A is some regularnon-negative matrix method. All the results can be transferred to the particular casesof the family Aα, e.g., to the families of some generalized Nörlund methods (N, pαn, qn)(see [4]).

The research was partially supported by Estonian Targeted Financing Project SF140083s08.

REFERENCES

[1] E. Kolk, Matrix summability of statistically convergent sequences, Analysis 13(1993), 77–83.

[2] F. Móricz, Tauberian conditions under which statistical convergence follows fromstatistical summability (C, 1), J. Math. Anal. Appl. 275 (2002), 277–287.

121

[3] V. Soomer and A. Tali, On strong summability of sequences, Acta Comment.Univ. Tartu. Math. 11 (2007), 57–68.

[4] U. Stadtmüller and A. Tali, Comparison of certain summability methods byspeeds of convergence, Anal. Math. 29 (2003), 227–242.

[5] U. Stadtmüller and A. Tali, Strong summability in certain families of summa-bility methods, Acta Sci. Math. (Szeged) 70 (2004), 639–657.

Institute of Cybernetics, Tallinn University of Technology, 12618 Tallinn, 21 Akadeemiatee, EstoniaE-mail: [email protected] of Mathematics and Natural Sciences, Tallinn University, 10120 Tallinn, 25Narva mnt, EstoniaE-mail: [email protected]

122


TAUBERIAN REMAINDER THEOREMSFOR THE (N, p) SUMMABILITY METHOD

Sefa Anıl Sezer and İbrahim Çanak

In this study we obtain some Tauberian conditions to deduce x ∈ mλ from (N, p)x ∈mλ. The results generalize results proved by Meronen and Tammeraid [8].

REFERENCES

[1] İ. Çanak and Ü. Totur, Some Tauberian theorems for the weighted mean methodsof summability, Comput. Math. Appl. 62 (2011), 2609–2615.

[2] M. Dik, Tauberian theorems for sequences with moderately oscillatory control mod-uli, Math. Morav. 5 (2001), 57–94.

[3] G. Kangro, A Tauberian remainder theorem for the Riesz method, Tartu Riikl. Ül.Toimetised 277 (1971), 155–160.

[4] O. Meronen and I. Tammeraid, Generalized Euler-Knopp method and conver-gence acceleration, Math. Model. Anal. 11 (2006), 87–94.

[5] O. Meronen and I. Tammeraid, Generalized Nörlund method and convergenceacceleration, Math. Model. Anal. 12 (2007), 195–204.

[6] O. Meronen and I. Tammeraid, Generalized linear methods and gap Tauberianremainder theorems, Math. Model. Anal. 13 (2008), 223–232.

[7] O. Meronen and I. Tammeraid, Several theorems on λ-summable series, Math.Model. Anal. 15 (2010), 97–102.

[8] O. Meronen and I. Tammeraid, General control modulo and Tauberian remaindertheorems for (C, 1) summability, Math. Model. Anal. 18 (2013), 97–102.

[9] A. Šeletski and A. Tali, Comparison of speeds of convergence in Riesz-type fam-ilies of summability methods. II., Math. Model. Anal. 15 (2010), 103–112.

[10] I. Tammeraid, Tauberian theorems with a remainder term for the Cesàro andHölder summability methods, Tartu Riikl. Ül. Toimetised 277 (1971), 161–170.

Department of Mathematics, Ege University, 35100 İzmir, TurkeyE-mails: [email protected], [email protected] of Mathematics, İstanbul Medeniyet University, 34720 İstanbul, TurkeyE-mail: [email protected]

123


AROUND SAMPLING THEOREM

Maria Skopina

The well-known sampling theorem states that

f(x) =∑

n∈Zf(2−jn)

sinπ(2jx− n)

π(2jx− n)(3)

for any function f ∈ L2(R) whose Fourier transform is supported on [−2j−1, 2j−1]. Fromthe point of view of wavelet theory, (3) is not a theorem, it is just an illustration for theShannon MRA. Indeed, the function φ(x) = sinπx

πx is a scaling function for this MRA, anda function f belongs to the sample space Vj if and only if its Fourier transform is sup-ported on [−2j−1, 2j−1]. So, such a function f can be expanded as f =

∑n∈Z〈f, φjn〉φjn,

where φjn(x) = 2j/2φ(2jx+n), which coincides with (3). Also, since Vjj∈Z is an MRA,any f ∈ L2(R) can be represented as

f = limj→+∞

∑

n∈Z〈f, φjn〉φjn. (4)

Moreover, (4) has an arbitrary large approximation order. This happens because thefunction φ(x) = sinπx

πx is band-limited. A similar property cannot be valid for othernatural classes of φ, in particular, for compactly supported φ.

We study operators Qjf =∑

n∈Z〈f, φjn〉φjn for a class of band-limited functions φand a wide class of tempered distributions φ. Convergence of Qjf to f as j → +∞ in L2-norm is proved under a very mild assumption on φ, φ, and the rate of convergence is equalto the order of Strang–Fix condition for φ. To study convergence in Lp, p > 1, we assume

that there exists δ ∈ (0, 1/2) such that φφ = 1 a.e. on [−δ, δ], φ = 0 a.e. on [l − δ, l + δ]for all l ∈ Z \ 0. For appropriate band-limited or compactly supported functionsφ, the estimate ‖f −Qjf‖p ≤ Cωr(f, 2

−j)Lp , where ωr denotes the r-th modulus ofcontinuity, is obtained for arbitrary r ∈ N. For tempered distributions φ, we provethat Qjf tends to f , f ∈ S, in Lp-norm, p ≥ 2, with an arbitrary large approximationorder. In particular, for some class of differential operators L, we consider φ such thatQjf =

∑n∈Z Lf(2

−j ·)(n)φjn. The corresponding wavelet frame-type expansions arefound.

St. Petersburg State UniversityE-mail: [email protected]

124


ON THE BEST APPROXIMATIONOF FUNCTIONS FROM HÖLDER CLASSESBY A SUBSETOF LINEAR FINITE-RANK POSITIVE METHODS

Dmytro Skorokhodov

Let C be the space of continuous functions on [0, 1] endowed with the standard norm‖f‖ = max|f(x)| : x ∈ [0, 1], and let ω be a concave modulus of continuity, i.e.,a concave, non-decreasing function satisfying ω(0) = 0. Let Hω denote the class offunctions f ∈ C such that |f(x)− f(y)| 6 ω(|x − y|), x, y ∈ [0, 1]. In particular, whenω(t) = tα, α ∈ (0, 1], the classes Hω are called the Hölder classes and are denoted Hα.

For N ∈ N, let LN be the set of all linear continuous N -rank operators A : C → C.We consider the problem of the best approximation of functions from Hω by methodsfrom some subset M ⊂ LN . To this end we define

λN (Hω;M) := infA∈N

supf∈Hω

‖f −Af‖ .

Let us remark that the quantity λN (Hω;LN ) is called the linear width of class Hω inspace C. The question of its evaluation remains open and was repeatedly posed byN. P. Korneichuk (see, for instance, [1]). We solve this problem for N = 1.

In addition, we find λN (Hω;M) for the class M = L++N of linear continuous N -rank

operators A that can be represented in the form

Af = e1

∫ 1

0f(t) dg1(t) + . . .+ eN

∫ 1

0f(t) dgN (t), f ∈ C,

with some non-negative continuous functions e1, e2, . . . , eN and non-decreasing functionsg1, g2, . . . , gN . In particular, we obtain

λN(Hα;L++

N

)=λ1 (H

α;L1)

Nα=

Γ(2− α)Γ(12 + α

2

)

2NαΓ(32 − α

2

) .

REFERENCES

[1] N. P. Korneichuk, Extremal values of functionals and the best approximationon classes of periodic functions (in Russian), Izv. Akad. Nauk SSSR Ser. Mat. 35(1971), 93–124.

125

Department of Mechanics and Mathematics, Dnepropetrovsk National University, 49010Dnepropetrovsk, 72 Gagarina avenue, UkraineE-mail: [email protected]

126


REMARKS ON RANK FUNCTIONSAND RANK VARIETIES

Marcin Skrzyński

A function ρ : N −→ N is said to be a rank function if it is weakly decreasing andsuch that

∀ j ∈ N \ 0 : ρ(j − 1) + ρ(j + 1) ≥ 2ρ(j).

Let Mn(F) be the vector space of all the n × n matrices over a field F. One can provethat ρ : N −→ N is a rank function if and only if

∃A ∈ Mρ(0)(F) ∀ j ∈ N : rank(Aj) = ρ(j).

(In the sequel we write rA(j) instead of rank(Aj)). The pointwise inequality is a partialorder on the set of all the rank functions.

It can be shown that if the Zariski closure of a set E ⊆ Mn(F) is irreducible, then theset of rank functions rA : A ∈ E has the greatest element [3]. If ρ is a rank function,then

Xρ = A ∈ Mρ(0)(F) : rA ≤ ρis an algebraic set of matrices, referred to as a rank variety [1]. Rank functions alsoappear in the Gerstenhaber–Hesselink theorem on the closure of a nilpotent orbit.

In the talk, we will present some new and some older results on rank functions andtheir applications in matrix theory and algebraic geometry.

REFERENCES

[1] D. Eisenbud and D. Saltman, Rank varieties of matrices, Commutative alge-bra, Proc. Microprogram, Berkeley/CA (USA) 1989, Math. Sci. Res. Inst. Publ. 15(1989), 173–212.

[2] P. Pokora and M. Skrzyński, Rank function equations, Ann. Univ. Paedagog.Crac. Stud. Math. 11 (2012), 101–109.

[3] M. Skrzyński, Remarks on applications of rank functions to algebraic sets of ma-trices, Demonstr. Math. 32 (1999), 263–271.

[4] M. Skrzyński, Irreducible algebraic sets of matrices with dominant restriction ofthe characteristic map, Math. Bohem. 128 (2003), 91–101.

127

Institute of Mathematics, Cracow University of Technology, ul. Warszawska 24, 31-155Kraków, PolandE-mail: [email protected]

128


EPIMORPHISMS IN CERTAIN CATEGORIESOF PARTIALLY ORDERED SEMIGROUPS

Nasir Sohail

A partially ordered semigroup (briefly, posemigroup) is a semigroup S endowed with apartial order ≤ that is compatible with the binary operation (i.e. for all s1, s2, t1, t2 ∈ S,(s1 ≤ t1, s2 ≤ t2) implies s1s2 ≤ t1t2). A posemigroup homomorphism f : S −→ T isa monotone semigroup homomorphism (i.e. for all s1, s2 ∈ S, f(s1s2) = f(s1)f(s2) ands1 ≤ s2 in S implies f(s1) ≤ f(s2) in T ). One can easily observe that f is necessarilyan epimorphism in the category of all posemigroups if it is such in the category of allsemigroups (where in the latter case we simply disregarded the orders). We show that theconverse of this statement, which may not be true in general, holds in certain categoriesof posemigroups (equivalently, semigroups).

REFERENCES

[1] N. Sohail, Zigzag theorem for partially ordered monoids, Comm. Algebra (to ap-pear).

[2] N. Sohail, Absolute closure for pomonoids, submitted.

Institute of Mathematics, University of Tartu, EstoniaE-mail: [email protected]

129


IDEALS WITH AT MOST COUNTABLE HULLIN CERTAIN ALGEBRAS OF ANALYTIC FUNCTIONS

Andrzej Sołtysiak

Closed ideals of subalgebras of the classical disc algebra A(D) were investigated bymany authors. We present an extension of a result of Agrafeuil and Zarabi [1] (andalso Faıvyševskiı [2, 3]) showing that under certain natural assumptions and a modifiedDitkin’s condition every closed ideal with at most countable hull of the given algebra Bis standard.

Next using this result we describe closed ideals with at most countable hull in algebrasA(α)(C+) (α > 0) of bounded analytic functions on the right half-plane satisfying certainconditions on the boundary.

The talk is based on the joint work [4,5] with Antoni Wawrzyńczyk from UniversidadAutónoma Metropolitana-Iztapalapa in México City.

REFERENCES

[1] C. Agrafeuil and M. Zarrabi, Closed ideals with countable hull in algebras ofanalytic functions smooth up to the boundary, Publ. Mat. 52 (2008), 19–56.

[2] V. M. Faıvyševskiı, The structure of the ideals of certain algebras of analyticfunctions, (Russian), Dokl. Akad. Nauk SSSR 211 (1973), 537–539; translation in:Soviet Math. Dokl. 14 (1973), 1067–1070.

[3] V. M. Faıvyševskiı, Spectral synthesis in Banach algebras of functions analytic inthe disc, (Russian), Funktsional. Anal. i Priložen. 8 (1974), 85–86; translation in:Functional Anal. Appl. 8 (1974), 268–269.

[4] A. Sołtysiak and A. Wawrzyńczyk, Ditkin’s condition and ideals with at mostcountable hull in algebras of functions analytic in the unit disc, Comment. Math. 52(2012), 101–112.

[5] A. Sołtysiak and A. Wawrzyńczyk, Ideals with at most countable hull in certainalgebras of functions analytic on the half-plane, Bol. Soc. Mat. Mexicana. 19 (2013),91–100.

Faculty of Mathematics and Computer Science, Adam Mickiewicz University, ul. Umul-towska 87, 61–614 Poznań, PolandE-mail: [email protected]

130


JOINT CONTINUITYVERSUS SEPARATE CONTINUITY:ON A CLASS OF NAMIOKA SPACES

Alexander Šostak

The problem of studying sets of points of joint continuity for a separately continuousmapping provoked interest of many mathematicians. Probably, the first one to be men-tioned here is R. Baire (1899). In more recent times important progress in studying thisproblem was done by I. Namioka (1974) who found some conditions on spaces X allowingto conclude that if f : X × Y →M is separately continuous, where M is a metric spaceand X is compact, then there exists a dense Gδ subset A of X such that f : A×Y →Mis jointly continuous at each point of A × Y . Following J. P. R. Christensen (1981) aspace X is called a Namioka space if the above property holds for any metric space Mand any compact space Y . The problem of extending known classes of Namioka spaceswas studied by many authors. In particular, J. Saint Raymond (1983) proved that everyseparable Baire space is Namioka. In the same work he applied a topological game forthe study of Namioka spaces.

In our talk we propose an alternative topological game and use it to characterize aclass of Namioka spaces.

In collaboration with Andrzej Szymanski, University of Slippery Rock, USA.

Zel,l,u street 8, Riga LV-1002, LatviaE-mail: [email protected]

131


GROWTH RATES OF FINITE ALGEBRAS

Ágnes Szendrei

For a finite algebra A and a positive integer n, what is the minimum number ofelements needed to generate An? If dA(n) represents this number, then dA(n) goes toinfinity monotonically as n goes to infinity. I will talk about how the structure of Ainfluences the rate at which dA(n) increases.

Department of Mathematics, University of Colorado, Boulder CO 80309-0395E-mail: [email protected]

132


APPROXIMATION ERROR OFGENERALIZED SHANNON SAMPLING OPERATORSWITH BANDLIMITED KERNELSIN TERMSOF AN AVERAGED MODULUS OF SMOOTHNESS

Gert Tamberg

The generalized sampling operator is given by (t ∈ R; w > 0)

(Swf)(t) :=∞∑

k=−∞f(k

w)s(wt− k). (1)

In this talk we study an even band-limited kernel s, defined as Fourier cosine transformof an even window function λ ∈ C[−1,1], λ(0) = 1, λ(u) = 0 (|u| > 1).

We will estimate the order of approximation of the sampling operator (1) for functionsf belonging to a suitable subspace Λp ⊂ Lp(R) (see also [1]) in terms of an averagedmodulus of smoothness τ2r (see [2]).Theorem. Let sampling operator Srw (w > 0) be defined by the kernel s with λ = λr andfor some r ∈ N let

λr(u) := 1−∞∑

j=r

cju2j ,

∞∑

j=r

|cj | 6 ∞.

Then for f ∈ Λp (1 6 p <∞)

‖SrW f − f‖p 6Mrτ2r(f ;1

w)p.

The constants Mr are independent of f and w.This research was partially supported by Estonian Science Foundation Grant 9383

and Estonian Targeted Financing Project SF0140011s09.

REFERENCES

[1] P. L. Butzer, C. Bardaro, R. L. Stens, and G. Vinti, Approximation errorof the Whittaker cardinal series in terms of an averaged modulus of smoothnesscovering discontinuous signals, J. Math. Anal. Appl. 316 (2006), 269–306.

[2] B. Sendov and V. Popov, The Averaged Moduli of Smoothness, Wiley, Chichester,1988.

133

Dept. of Mathematics, Tallinn University of Technology, 19823 Tallinn, Ehitajate tee 5,EstoniaE-mail: [email protected]

134


ALGEBRAIC COSINE VALUESAT RATIONAL MULTIPLES OF π

Vichian Laohakosol and Pinthira Tangsupphathawat

Rational and higher algebraic values of the cosine function have been of much interestfor quite some time, cf. [2, 3, 5, 6]. As early as 1933, D. H. Lehmer, [3], proved thatif k/n, n > 2, is an irreducible fraction, then 2 cos(2πk/n) is an algebraic integer ofdegree ϕ(n)/2, where ϕ(n) is the Euler’s totient function. Lehmer’s proof makes use ofcyclotomic polynomials. As a consequence, we have ( [4, Theorem 6.16, pp. 308–309]):let θ = rπ be a rational multiple of π, then cos θ is irrational except when cos θ =0,±1/2,±1. Recently, Varona, [6], proved that if r ∈ Q ∩ [0, 1], then arccos(

√r) is a

rational multiple of π if and only if r ∈ 0, 1/4, 1/2, 3/4, 1. His proof is elementary andis similar to the proof of [1, Theorem 4, p. 32].

Our objectives here are:• to use elementary trigonometric identities to find all nonnegative rational and some

quadratic values of the cosine function at rational multiples of π extending the resultin [6];

• to use Lehmer’s results in [3] to determine all algebraic cosine values at rationalmultiples of π;

• to explicitly work out all algebraic cosine values of degree less than 5 at rationalmultiples of π.

The research was partially supported by the Thailand Research Fund through theRoyal Golden Jubilee Ph.D. Program (Grant No. PHD/0022/2552).

REFERENCES

[1] M. Aigner and G. M. Ziegler, Proofs from THE BOOK, Springer, Berlin, 1998.

[2] R. W. Hamming, The transcendental character of cosx, Amer. Math. Monthly 52(1945), 336–337.

[3] D. H. Lehmer, A note on trigonometric algebraic numbers, Amer. Math. Monthly40 (1933) pp. 165–166.

[4] I. Niven, H.S. Zuckerman, and H. L. Montgomery, An Introduction to theTheory of Numbers, 5th ed., Wiley, New York, 1991.

[5] J. M. H. Olmsted, Rational values of trigonometric functions, Amer. Math.Monthly 52 (1945), 507–508.

135

[6] J. L. Varona, Rational values of the arccosine function, Central European J. Math.4 (2006), 319–322.

Department of Mathematics, Kasetsart University, Bangkok 10900, ThailandE-mails: [email protected], [email protected]

136


SOME DUALITY RESULTSON BOUNDED APPROXIMATION PROPERTIESOF PAIRS

Eve Oja and Silja Treialt

The bounded approximation property for pairs (X,Y ) consisting of a Banach space Xand a fixed subspace Y was recently introduced by Figiel, Johnson, and Pełczyński [FJP].

Johnson [J] proved that if the dual space X∗ of a Banach space X has the boundedapproximation property, then it also has the bounded duality approximation property.We extend Johnson’s result to the pairs of Banach spaces as follows. The pair (X∗, Y ⊥)has the bounded approximation property if and only if the pair (X,Y ) has the boundedduality approximation property.

We also present several reformulations of the bounded approximation property ofpairs and its duality version, and study possibilities for lifting the bounded approximationproperties of a pair (X,Y ) to the pair (X∗, Y ⊥) in some special cases.


REFERENCES

[FJP] T. Figiel, W. B. Johnson, and A. Pełczyński, Some approximation propertiesof Banach spaces and Banach lattices, Israel J. Math. 183 (2011), 199–231.

[J] W. B. Johnson, On the existence of strongly series summable Markushevich basesin Banach spaces, Trans. Amer. Math. Soc. 157 (1971), 481–486.

Faculty of Mathematics and Computer Science, University of Tartu, 50409 Tartu, J. Liivi2, Estonia; Estonian Academy of Sciences, 10130 Tallinn, Kohtu 6, EstoniaE-mail: [email protected] of Mathematics and Computer Science, University of Tartu, 50409 Tartu, J. Liivi2, EstoniaE-mail: [email protected]

137


THE KA- AND THE KA-UNIFORMAPPROXIMATION PROPERTY

Silvia Lassalle and Pablo Turco

In 1984, Carl and Stephani defined, for a fixed Banach operator ideal A, the notion ofA-compact sets and the operator ideal of A-compact operators, denoted by KA. We usethe Carl and Stephani theory to inspect two types of approximation properties. The firstis rather standard. We say that a Banach space E has the KA-uniform approximationproperty if the identity map is uniformly approximated by finite rank operators on A-compact sets. For the second one, we introduce a way to measure the size of A-compactsets and use it to give a norm on KA. The geometric results obtained for KA are appliedto give different characterizations of the KA-approximation property, defined by Oja andthe authors independently. This approach allows us to undertake the study of bothapproximation properties in tandem. In particular, when A = N p, the ideal of rightp-nuclear operators, we cover the p-approximation property and the κp-approximationproperty, which were studied in the past ten years by several authors.

The results of this talk are contained in a joint work [1] with S. Lassalle.This project was supported in part by UBACyT Grant 1-746, CONICET-PIP 0624

and ANPCyT PICT 2011-1456.

REFERENCES

[1] S. Lassalle and P. Turco, The Banach ideal of A-compact operatorsand related approximation properties, J. Funct. Anal (2013), to appear,http://dx.doi.org/10.1016/.j.jfa.2013.07.001

Departamento de Matemática, Universidad de San Andrés, Vito Dumas 284, (B1644BID)Victoria, Buenos Aires, Argentina and IMAS - CONICET.E-mail: [email protected] de Matemática - Pab I, Facultad de Cs. Exactas y Naturales, Universidadde Buenos Aires, (1428) Buenos Aires, Argentina and IMAS - CONICETE-mail: [email protected]

138


SURVEY OF VECTOR-VALUEDANALYTIC COMPOSITION OPERATORS

Hans-Olav Tylli

I will survey some recent results, due to a number of authors, and problems concerningvector-valued analytic composition operators first considered in [LST].

Let ϕ : D → D be an analytic map, where D is the unit disk in C. As an extension ofthe classical scalar-valued theory of composition operators it is of interest to study the op-erators f 7→ Cϕ(f) = f ϕ on various Banach spaces consisting of vector-valued analyticfunctions f : D → X, where X is an infinite dimensional complex Banach space. Rele-vant examples include the vector-valued Hardy spaces Hp(X), Bergman spaces Ap(X),spaces BMOA(X) of functions of bounded mean oscillation, as well as weak versions ofthese spaces.

I will focus on properties illustrating some of the similarities, differences, as well asnew phenomena encountered in comparison to the classical scalar-valued setting. Forinstance, weak compactness is often the smallest relevant qualitative property, whileone may also consider composition operators from weak to strong type spaces. Openproblems motivated by the vector-valued setting will be stated.

REFERENCES

[LST] P.D. Liu, E. Saksman, and H-O. Tylli, Small composition operators on ana-lytic vector-valued function spaces, Pacific J. M. 184 (1998), 295–310.

Department of Mathematics and Statistics, University of Helsinki, FinlandE-mail: [email protected]

139


ERROR ESTIMATESFOR CARDINAL SPLINE INTERPOLATIONAND QUASI-INTERPOLATION

Gennadi Vainikko

Denote by Sh,m =fh ∈ Cm−2 : fh

∣∣∣[ih,(i+1)h]

∈ Pm−1, i ∈ Z

the space of (cardinal)

splines of step size h > 0 and degree m − 1, m ∈ N. For f ∈ BC(R), there exists aunique bounded (Wiener-Schoenberg) interpolant Qh,mf ∈ Sh,m interpolating f at thepoints (k + m

2 )h, k ∈ Z. It holds [1] that

supf∈Wm,∞(R), ‖f (m)‖∞=1

‖ f −Qh,mf ‖∞= κm+1π−mhm,

where κm = 4π

∑∞k=0(−1)km(2k + 1)−m is the Favard constant. There exists no approxi-

mation method of higher accuracy provided that only the values f((k+ m2 )h), k ∈ Z, are

exploited by the method.For m ≥ 3, computation of (Qh,mf)(x) at an intermediate point x ∈ R needs all the

values f((k+ m2 )h), k ∈ Z. We introduce a quasi-interpolant Q′

h,mf ∈ Sh,m, “almost” [2]preserving the accuracy of Qh,mf and such that (Q′

h,mf)(x) needs the values of f onlyat (k + m

2 )h ∈ [x−mh, x+mh].Due to the last property, quasi-interpolation is preferable while designing [3] fully

discrete methods for integral equations; main ideas will be explained but we do not gointo details in this talk.

REFERENCES

[1] Gennadi Vainikko, Error estimates for the cardinal spline interpolation, J. Anal.Appl. (ZAA) 28 (2009), 205–222.

[2] Evely Leetma and Gennadi Vainikko, Quasi-interpolation by splines on theuniform knot sets, Math. Model. Anal. 12 (2007), 107–120.

[3] Eero Vainikko and Gennadi Vainikko, Product quasi-interpolation in logarith-mically singular integral equations, Math. Model. Anal 17 (2012), 696–714.

Institute of Mathematics, University of Tartu, 50409 Tartu, J. Liivi 2, EstoniaE-mail: [email protected]

140


RESTRICTED REVERSIBLE RINGS

Stefan Veldsman

Reversibility of rings is a generalization of commutativity, but more-than-often thisweaker commutativity is a consequence of the absence of certain zero products. Forexample, a reversible ring is prime if and only if it is an integral domain and a ring isreduced if and only if it is reversible and semiprime. Here we define and investigate classesof more restricted reversible rings which fulfill stronger commutativity requirements; forexample, rings R for which ab = 0 = ac+ db implies ba = 0 = ca+ bd.

Department of Mathematics, Sultan Qaboos University, Muscat, OmanE-mail: [email protected]

141


MATRIX IDENTITIES INVOLVINGMULTIPLICATION AND TRANSPOSITION

Mikhail Volkov

Matrices and matrix operations constitute basic tools for algebra and analysis. Im-portant properties of matrix operations are often expressed in form of laws or identitiessuch as the associative law for multiplication of matrices. Studying matrix identities thatinvolve multiplication and addition is a classic research direction motivated by severalimportant problems in geometry and algebra. Matrix identities involving along withmultiplication and addition also certain involution operations (such as taking the usualor symplectic transpose of a matrix) have attracted much attention as well.

If one aims to classify matrix identities of a certain type, then a natural approachis to look for a collection of “basic” identities such that all other identities would followfrom these basic identities. Such a collection is usually referred to as a basis. Forinstance, all identities of matrices over an infinite field involving only multiplication areknown to follow from the associative law. Thus, the associative law forms a basis ofsuch “multiplicative” identities. For identities of matrices over a finite field or a fieldof characteristic 0 involving both multiplication and addition, the powerful results byKruse–L’vov and Kemer ensure the existence of a finite basis. In contrast, multiplicativeidentities of matrices over a finite field admit no finite basis.

Here we consider matrix identities involving multiplication and one or two naturalone-place operations such as taking various transposes or Moore–Penrose inversion. Ourresults may be summarized as follows.

None of the following sets of matrix identities admits a finite basis:

• the identities of n×n-matrices over a finite field involving multiplication and usualtransposition;

• the identities of 2n × 2n-matrices over a finite field involving multiplication andsymplectic transposition;

• the identities of 2 × 2-matrices over the field of complex numbers involving ei-ther multiplication and Moore–Penrose inversion or multiplication, Moore–Penroseinversion and Hermitian conjugation;

• the identities of Boolean n×n-matrices involving multiplication and transposition.

Institute of Mathematics and Computer Science, Ural Federal University, Lenina 51,620000 Ekaterinburg, RussiaE-mail: [email protected]

142


ON THE HAHN PROPERTY OF BOUNDED DOMAINSOF SPECIAL MATRIX METHODS

Maria Zeltser

We discuss 3 properties of sequence spaces characterizing to which extent sequencespaces are determined by the sequences of 0’s and 1’s that they contain: the Hahnproperty, the separable Hahn property, and the matrix Hahn property (cf. [1]). TheHahn property is stronger than the separable Hahn property and the latter one is strongerthan the matrix Hahn property. One simple necessary condition for all Hahn propertiesis that the beta dual of the set of sequences contained in a sequence space E coincideswith the beta dual of E. Generally this assumption is not sufficient even for the matrixHahn property. However, if we ask additionally E to be a solid sequence space with thebeta dual equal to ℓ1, then E has the separable Hahn property (cf. [2], Theorem 6). Wewill see that generally even these additional assumptions do not imply that E has theHahn property. However, for some classes of sequence spaces E this implication holds;for example, if E is the bounded domain of a regular Riesz or Hausdorff method.

In case of regular Riesz and Hausdorff matrices the bounded summability domain hasthe Hahn property if and only if the matrix has spreading rows. Relying on Boos–Leigermethods (cf. [3]) we will show that this condition also implies the Hahn property of thebounded summability domain in case of a generalized Riesz matrix obtained as a rowsubmatrix of a Riesz matrix.

This research was supported by the Estonian Science Foundation Grant 8627, Eu-ropean Regional Development Fund (Centre of Excellence “Mesosystems: Theory andApplications”, TK114) and Estonian Targeted Financing Project SF0130010s12.

REFERENCES

[1] G. Bennett, J. Boos, and T. Leiger, Sequences of 0’s and 1’s, Studia Math.149 (2002), 75–99.

[2] J. Boos and T. Leiger, Addendum: Sequences of 0’s and 1’s [Studia Math. 149(1) (2002), 75–99.], Studia Math. 171 (2005), 305–309.

[3] J. Boos and T. Leiger, Strongly nonatomic densities defined by certain matrices,Math. Slovaca 63 (2013), 573–586.

Department of Mathematics, Tallinn University, 10120 Tallinn, Narva mnt. 25, EstoniaE-mail: [email protected]

143


INEQUALITIESOF THE GENERALIZED JACKSON THEOREM TYPEFOR BEST APPROXIMATIONS

Vladimir Zhuk

Let C be the space of 2π-periodic continuous functions, equipped with the uniformnorm, and let E(f) be the best approximation of function f by trigonometric polynomialsof degree less than or equal to n in the space C. It is well known that for all f ∈ C andn ∈ Z+ the following inequality holds

En(f) ≤ C(r, γ)ωr

(f,

γr

n+ 1

), (1)

where C depends only on its arguments, ωr(f, ·) denotes the modulus of continuity of or-der r for f in the space C. Inequalities of such kind play important role in approximationtheory, and a large number of papers is devoted to their study in different directions.Inequalities like (1) are called direct theorems of approximation theory or generalizedJackson inequalities. There is a number of known approaches to establish such inequal-ities. We suggest very simple methods (with a lot of applications at the same time),which allow to establish inequalities analogous to generalized Jackson theorem for bestapproximations.

Department of Applied Mathematics and Control Processes, Saint Petersburg State Uni-versity, 198504 St. Petersburg, 35 Universitetskii pr., RussiaE-mail: [email protected]

144


THE ASYMPTOTICALLY COMMUTINGBOUNDED APPROXIMATION PROPERTYOF BANACH SPACES

Eve Oja and Indrek Zolk

Our departure point is the following two theorems of Nigel J. Kalton et al. (see,e.g., [C, Theorems 4.6 and 9.3]):

1) separable Banach spaces with the metric approximation property (AP) have thecommuting metric AP (Casazza–Kalton);

2) in the non-separable case, the commuting bounded AP implies the separablecomplementation property, hence, e.g., ℓ∞ fails the commuting bounded AP (Casazza–Kalton–Wojtaszczyk).

We introduce and study a (strict) weakening of the commuting bounded AP – theasymptotically commuting bounded AP. It turns out that any dual space with thebounded AP also enjoys this property. Our principal result is that a Banach spacewith the asymptotically commuting bounded AP is saturated with separable closed sub-spaces having a nice form of the commuting bounded AP. These subspaces are locallycomplemented and, in some special cases, even complemented.


REFERENCES

[C] P. G. Casazza, Approximation properties. In: W.B. Johnson and J. Lindenstrauss(eds.) Handbook of the Geometry of Banach Spaces. Volume 1, Elsevier (2001),271–316.

Institute of Mathematics, University of Tartu, 50409 Tartu, J. Liivi 2, Estonia; EstonianAcademy of Sciences, 10130 Tallinn, Kohtu 6, EstoniaE-mail: [email protected] of Mathematics, University of Tartu, 50409 Tartu, J. Liivi 2, EstoniaE-mail: [email protected]

145

146

Author index

Aasma, Ants, 42Abel, Mati, 108Abrahamsen, Trond A., 43Ain, Kati, 44Akduman, Setenay, 45Aron, Richard M., 46

Bendová, Hana, 47Bloshanskaya, Svetlana, 85Bloshanskii, Igor, 63, 85Boos, Johann, 48

Cırulis, Janis, 56del Campo, Ricardo, 49, 95, 100Çanak, İbrahim, 51, 123Cascales, Bernardo, 53Ciaś, Tomasz, 54

Dales, Harold Garth, 57Delgado, Juan Manuel, 58Diestel, Joe, 59Dudzik, Dariusz, 60

Fernández, Antonio, 49, 61, 95, 100

Godefroy, Gilles, 62Grafov, Denis, 63Grygiel, Joanna, 65Grygiel, Katarzyna, 65

Hájek, Petr, 66Haller, Rainis, 83Henno, Jaak, 67

Israfılov, Daniyal, 70

Janelidze, George, 72Jiménez Fernández, Eduardo, 73

Johanson, Marje, 74Johnson, William, 75Juan, María Aranzazu, 73

Kaarli, Kalle, 39, 76Kearnes, Keith A., 77Kivinukk, Andi, 78, 118Košik, Oleg, 79Kuznetsova, Olga I., 80

Laan, Valdis, 82Langemets, Johann, 43, 83Laohakosol, Vichian, 84, 135Lassalle, Silvia, 138Leiger, Toivo, 39Lifantseva, Olga, 85Liflyand, Elijah, 87Lillemets, Rauni, 44, 89Lima, Vegard, 43Lindström, Mikael, 90

Machida, Hajime, 91Malkowsky, Eberhard, 92Manzano, Antonio, 49, 95Márki, László, 93Mathieu, Martin, 94Mayoral, Fernando, 49, 61, 95, 100Mayr, Peter, 79, 97Metsmägi, Tarmo, 98Montesinos, Vicente, 99

Naranjo, Francisco, 49, 61, 95, 100Niglas, Heiki, 101Novikova, Anna, 102Nygaard, Olav, 103

Oja, Eve, 44, 101, 137, 145

147

Orlova, Olga, 104

Pérez-Tiscareño, Reyna María, 108Põldvere, Märt, 83Pavlyk, Kateryna, 106Pellegrino, Daniel, 107Piñeiro, Cándido, 58Pietsch, Albrecht, 109Pintoptang, Umarin, 110Plichko, Anatolij, 111Podkorytov, Anatolii N., 80Przestacki, Adam, 112

Rack, Heinz-Joachim, 113Reinov, Oleg I., 115Rozhinskaya, Ksenia, 116Rueda, Pilar, 117

Saksa, Anna, 118Sánchez-Pérez, Enrique A., 73, 119Savas, Ekrem, 120Schlumprecht, Thomas, 66Šeletski, Anna, 121Sezer, Sefa Anıl, 123Skopina, Maria, 124Skorokhodov, Dmytro, 125Skrzyński, Marcin, 127Sohail, Nasir, 129Sołtysiak, Andrzej, 130Šostak, Alexander, 131Szendrei, Ágnes, 132

Tali, Anne, 39, 121Tamberg, Gert, 104, 133Tangsupphathawat, Pinthira, 135Totur, Ümit, 51Treialt, Silja, 137Turco, Pablo, 138Tylli, Hans-Olav, 139

Vainikko, Gennadi, 140Veldsman, Stefan, 141Volkov, Mikhail, 142

Zeltser, Maria, 143

Zhuk, Vladimir, 144Zolk, Indrek, 101, 116, 145

148

Notes

149

Sun

8.00

8.45 8.45

9.00 9.00 9.00

10.00 10.00 10.00

10.50 10.50 10.50

11.00

11.10 11.10 11.10

12.00 12.00 12.00

13.00

13.00 13.00

13.30

13.35

13.50

14.00

14.10

14.25

14.30

14.40

15.00 15.00 15.00

15.10

15.35 15.30 Kivinukk

15.40 Aasma

16.00

16.05

16.10 Ain

16.30

16.35

16.40

17.00

17.05

17.35 Metsmägi

18.00 18.00

19.00

Monday Tuesday Wednesday Thursday Friday

Registration

Welcome to Estonia W. to Estonia Welcome to Estonia W. to Estonia

Opening; A. Tali Pietsch Diestel Dales JohnsonAron Volkov Szendrei Kearnes Janelidze

Coffee break Coffee break Coffee break Coffee break

Wal

kin

g(T

alli

nn)

McKenzie Boos Cascales GodefroyLunch break Lunch break Lunch break Closing

Analysis I(Lektoorium)

Algebra /An. I (Laika)

Analysis II(Apollo)

Johanson, Košik, Langemets,Lillemets, Niglas, Novikova,

Orlova, Rozhinskaya, Saksa,Treialt, Zolk

Excursions Analysis I(Lektoorium)


Analysis II(Apollo)

Hajek Márki Skopina Fernández Tangsupp-hathawat

Zeltser

Analysis I(Lektoorium)


Analysis II(Apollo)

Bendová Pavlyk Rack

Mathieu Mayr Malkowsky

Tylli Machida Liflyand

Nygaard Sohail Šeletski

Pellegrino Grygiel JiménezFernández Montesinos Veldsman Vainikko

Abrahamsen Laan Dudzik

Coffee break

Reg

istr

atio

n Plichko Cīrulis Savas Coffee break

Sołtysiak Skrzyński Kuznetsova

Coffee break Naranjo Laohakosol

Sánchez-Pérez

PérezTiscareño Israfılov Pintoptang Lifantseva

Lindström Kaarli Zhuk

Ciaś Çanak

Šostak Henno Grafov

Rueda del Campo Skoro-khodovDelgado Przestacki Sezer

Walking(Tallinn)Reinov Mayoral Tamberg

Turco Akduman

Wel

com

ere

cepti

on Walking excursion (Tartu) The ethics of publishing

and open access(H. G. Dales; open discussion)

Conference banquet

Documents

Kangro-100kangro100.ut.ee/kangro100-abstract_book.pdf · Contents Sponsors and partners 7 Scienti c and organizing committee 9 Plenary speakers 10 Programs 11 Program for participants