Classical and Modern Methods

in

Summability

Johann BoosFachbereich Mathematik

FernUniversitat–GesamthochschuleHagen, Germany

assisted by

Peter CassDepartment of Mathematics

The University of Western OntarioLondon, Ontario, Canada

toChristiane

and toDaniela and Nicolas

Preface

In its broadest meaning, summability theory , or in short summability , isthe theory of the assignment of limits, which is fundamental in analysis,function theory, topology and functional analysis. For instance, we are in-terested in the assignment of limits in the case of• real or complex sequences (xn) for the limit process ‘n →∞ ’.• series (convergence of series),• sequences and series of functions like power series, Fourier series, etc.

(pointwise, uniform or compact convergence),• limit of a function at a point (continuity, continuous extension),• differentiation of functions,• integration of functions,• sequences and series in topological (vector) spaces.

Except for examples of applications, we deal in this book with summabilityin a narrower sense, that is, with the assignment of limits in the case ofreal or complex sequences by so-called summability methods which aremost often defined in this book by an infinite matrix (matrix method) orby a power series (power series method).

We aim to give a broad introduction to summability theory and developsome of its most important methodology. We distinguish between classical(‘hard’) methods and modern (‘soft’) methods which are essentially basedon analytical and functional analytic methods, respectively1.

The book is subdivided into three parts which are, roughly speaking,devoted to classical methods (Part I, Chapters 1–5), modern methods (PartII, Chapters 6–8) and the combination of classical and modern methods(Part III, Chapters 9–11). Concerning summability, the heart of Part I iscontained in Chapters 2–4 where we deal mainly with inclusion, Toeplitz–Silverman-type theorems, consistency theorems and the like. In Chapters 2and 3 we discuss matrix methods like Cesaro methods, Hausdorff methodsand others, and power series methods like the Abel method and the Borelmethod. Chapter 4 deals with Tauberian theorems for certain (classes of)summability methods. Some applications are given in Chapter 5.

In Part II we investigate the structure of the domains of matrix meth-ods, that is the set of all sequences which are ‘summable’ by the method in

1 This distinction comes from the fact that functional analysis is the younger theoryand should not be understood in the sense that classical analysis has nothing new tooffer.

viii Preface

question. We apply modern methods, that is functional analytic methods,to these matters in Chapter 8. The essential tool here is FK-space theorywhich is due mainly to K. Zeller and which we develop in outline in Chap-ter 7. All functional analytic tools beyond the basics, which are appliedin Part II or III, are discussed in Chapter 6. So in this sense the bookis self-contained. To give an impression of the advantages and elegance offunctional analytic methods of proof we prove anew the Toeplitz–Silvermantheorems that we proved earlier in Part I by classical methods. Whereasthe classical methods require detailed and technical arguments, the func-tional analytic methods shorten these arguments and provide new insightsinto the meaning of some of the conditions that arise in the theorems beingproved. The functional analytic methods, however, are not constructive andthis can be a disadvantage in some situations. Both points of view shouldbe appreciated for what each can offer.

In Part III we continue certain investigations of Part I and II, for exam-ple the ‘consistency examinations’, where we combine classical and modernmethods of proof. Often we can handle broad steps of the proofs withfunctional analytic methods but we must still perform the fine work byrelatively extensive and technical classical methods. Whereas in Chapters9 and 10 we deal with problems concerning the ‘consistency’ of matrixmethods which arise in summability and are solved by using the tools oftopological sequence spaces, in Chapter 11 we deal with problems in thistheory and solve them by the application of results from summability. Soin Parts II and III we show—by way of examples—the connection betweensummability and topological sequence spaces and demonstrate how eachfield helps the other.

The book is designed as a textbook for a variety of courses on summa-bility and its applications at the graduate level. Its study requires solidknowledge of calculus and linear algebra as well as a basic knowledge offunction theory and, concerning the applications, of numerical analysis.The study of Parts II and III requires very little knowledge of topologybut basic knowledge of functional analysis concerning metric and normedspaces. As already mentioned, functional analytic tools, beyond the basicones, are discussed in Chapter 6.

At this time no textbook or monograph is available that presentssummability and its applications in the breadth undertaken here. Recentdevelopments and applications to some related topics are discussed andopen questions of interest to current research are posed. Thus this bookshould be of interest to those conducting research in summability or topo-logical sequence spaces. Mathematicians working in areas bordering onsummability should find much of interest. This book should also serve wellas a textbook for a variety of graduate courses in summability and topo-logical sequence spaces and provides ample background to books or paperswhich present the subject in overview, such as, for example, Kangro’s paper

Preface ix

[123] and Zeller and Beekmann’s book [267] which, in spite of databases likeMathSciNet from the American Mathematical Society and ZMATH fromZentralblatt der Mathematik (etc.), retain their value and importance.

Hagen J. B.April 2000

Acknowledgements

This book arose from a course for distance learning, entitled ‘Limitierungs-theorie’ (Limitierungstheorie is the German term for summability theory)which was written by me with the assistance of Georgia Thurner, typesetby C. Maurer and S. Sikora, and revised by me with the aid of K.-E. Spreng.Some parts of Chapter 3 and 4 come from another summability course fordistance learning (cf. [227]) which was written by U. Stadtmuller.

H. Tietz and K. Zeller drew my attention to a recent paper (cf. [240]) inwhich they give a modification of Wielandt’s well-known elegant proof ofthe Hardy–Littlewood O–Tauberian theorems for the Abel method. Thisis an elementary proof and I decided to use the material of this paper forthe important part of Section 4.4. Similarly, W. Kratz gave—for use inthis book—an elementary proof of the O–Tauberian theorem for the Borelmethod which he adapted from a more general situation in a joint paperwith U. Stadtmuller (cf. [134]). I use this material in Section 4.5.

In the fall of 1998 I stayed for a couple of weeks at the University ofManitoba (Winnipeg, Canada) at the invitation of M. R. Parameswaran.We had fruitful discussions concerning ‘potent matrices’ and I wrote, inthis period, Section 2.9 and the corresponding part in Section 3.2.

The Summability/Functional Analysis Group in my department, con-sisting of W. Beekmann, K.-G. Große-Erdmann, K.-E. Spreng (and me)discussed many aspects of the work for my book. My doctoral student andcolleague D. Seydel spent a lot of time proof reading and gave many hintsand valuable suggestions. Sylvia Sikora supported me in typesetting thebook in LATEX2ε. She typed some parts of the book and made thousandsof corrections and changes with incredible patience and accuracy.

I thank all these friends and colleagues for their support.A special word of thanks goes to Peter Cass (The University of Western

Ontario in London (Ontario, Canada)). From the beginning he aided theproject with discussions and suggestions. He went very carefully throughthe whole book and gave a ‘nearly uncountable’ set of suggestions andcorrections concerning both the language and the mathematical contents.

Last but not least, thanks to my loving family—my wife Christiane,my daughter Daniela and my son Nicolas—who provided an atmosphere ofunerring support and patience to make possible the realization of such alarge project as this book.

J. B.

Contents

PART I CLASSICAL METHODS IN SUMMABILITYAND APPLICATIONS

1 Convergence and divergence 31.1 The early history of summability—the devil’s in-

vention 31.2 Summability methods: definition and examples 51.3 Questions and basic notions 201.4 Notes on Chapter 1 25

2 Matrix methods: basic classical theory 262.1 Dealing with infinite series 272.2 Dealing with infinite matrices 342.3 Conservative matrix methods 392.4 Coercive and strongly conservative matrix methods 512.5 Abundance within domains; factor sequences 612.6 Comparison and consistency theorems 752.7 Triangles of type M 822.8 The mean value property 872.9 Potent matrix methods 922.10 Notes on Chapter 2 97

3 Special summability methods 993.1 Cesaro and Holder methods 1003.2 Weighted means, Riesz methods 1123.3 Norlund methods 1263.4 Hausdorff methods 1363.5 Methods of function theoretical type 1523.6 Summability methods defined by power series 1573.7 Notes on Chapter 3 165

4 Tauberian theorems 1674.1 Tauberian theorems for Cesaro methods 1684.2 Tauberian theorems for Riesz methods 1784.3 Tauberian theorems for power series methods 1864.4 Hardy–Littlewood’s O-theorems for the Abel

method 191

xii Contents

4.5 Hardy–Littlewood’s O-theorem for the Borelmethod 196

4.6 Notes on Chapter 4 204

5 Application of matrix methods 2055.1 Boundary behaviour of power series 2065.2 Analytic continuation 2145.3 Numerical solution of systems of linear equations 2285.4 Fourier effectiveness of matrix methods 2445.5 Notes on Chapter 5 256

PART II FUNCTIONAL ANALYTIC METHODS INSUMMABILITY

6 Functional analytic basis 2616.1 Topological spaces 2626.2 Semi-metric spaces 2686.3 Semi-normed spaces, Banach spaces 2806.4 Locally convex spaces 2926.5 Continuous linear maps and the dual space of a

locally convex space 3066.6 Dual pairs and compatible topologies 3146.7 Frechet spaces 3276.8 Barrelled spaces 335

7 Topological sequence spaces: K- and FK-spaces 3387.1 Sequence spaces and their ζ-duals 3397.2 K-spaces 3497.3 FK-spaces 3597.4 Functional analytic proofs of some Toeplitz–

Silverman-type theorems 3687.5 The dual of FK-spaces 3757.6 Distinguished subspaces of FK-spaces 3857.7 Notes on Chapter 7 393

8 Matrix methods: structure of the domains 3968.1 Domains of matrix methods as FK-spaces 3978.2 Distinguished subspaces of domains 4068.3 Replaceability and µ-uniqueness of matrices 4198.4 Examples 4238.5 Bounded divergent sequences in the domain 4388.6 Consistency and perfectness 4438.7 Replaceability and invariance 4488.8 Notes on Chapter 8 454

Contents xiii

PART III COMBINING CLASSICAL ANDFUNCTIONAL ANALYTIC METHODS

9 Consistency of matrix methods 4599.1 Consistency and theorems of Mazur–Orlicz type 4609.2 µ-bounded sequences and domains 4759.3 µ-consistency and µ-comparison 4839.4 Singularities of matrices 500

10 Saks spaces and bounded domains 51510.1 Saks spaces and mixed topologies 51610.2 The Saks space m ∩WE 52210.3 A theorem of Mazur–Orlicz type 52710.4 b-comparison through quotient representations 52910.5 Notes on Chapter 10 536

11 Some aspects of topological sequence spaces 53811.1 An inclusion theorem 53811.2 Gliding hump and oscillating properties 54011.3 Theorems of Toeplitz–Silverman type via sectional

convergence and . . . 54611.4 Barrelled K-spaces 55011.5 The sequences of zeros and ones in a sequence space 55711.6 Notes on Chapters 9 and 11 561

Bibliography 563

Index 575

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Index

List of symbols

Miscellaneous∗ , 1290 < µk ↑ ∞ , 476< M > , 6< , 39Dn(v) , 246G(T ) , 331HK(A) , 492

KAn (v) , 247

Ln , 252LA

n , 250Mn , 236∆n , 186Γ , 105Σα , 103, 232ΣRF , 229χM , 95V , 215

≡ , 235

Kern f , 286Sgn , 474Sol Q , 560conv A , 470sgn , 40µ∗A , 454⊕ , 6πj , 296, 349∼ , 109ϕt , 254ak(f) , 244iX , 350px(t) , 158

vBj (z) , 235

Conditions(Σ1) , 519MK(A) , 87S1 , 504S2 , 511S3 , 513S4 , 513O , 168

OL , 168OR , 168o , 168(Sp) , 44(Sp0) , 45, 160(Zn) , 44(Zr) , 63(Zs) , 46(Zs1) , 47KG, 93KP, 93

Distinguished subspaces of K-spacesBX , 356B +

X , 386FX , 356F +

X , 386SX , 356WX , 356

Distinguished subspaces of domainsBA , 407FA , 407IA , 78, 407I int

A , 416LA , 407PA , 407SA , 407TA , 407WA , 407ΛA , 78, 407

Domains of summability methodsEA , 397CA1 , 10eCB1 , 15CPp , 158–165, 186–191ωA , 13bω bB , 229cA , 13c0A , 13bcs bB , 229mA , 42

Dual spacesX∗ , 284X ′ , 284Xf , 356

576 Index

Xα , 341Xββ , 39, 394Xβ , 39, 341Xγ , 341Xζζ , 342

Function spacesB(X) , 284B(X, Y ) , 284BV ([0, 1]) , 144C(R) , 294C2π , 244M(D) , 269C , 10, 158eC , 15Cp , 158Abb(Z, X) , 267Hom(X, Y ) , 284

Limit functionalsbB–P

, 229

Λ ⊥A , 78, 407

limA , 13lim sup , 51Lim, 16A1– lim , 10A– lim , 13B1– lim , 15F– lim , 17Pp– lim , 158lim, 6

Metrics, semi-metricsdY , 271d∞ , 270dω , 273dp , 269

Neighbourhood (systems)Kε(x) , 270Uε , 282Uε(x) , 270

Udε (x) , 270

Upε , 282, 293B , 293B(x) , 263, 293BP , 293U , 282, 294U(x) , 263, 264Uτ (x) , 263Up , 282Ueps(x) , 270

Udeps(x) , 270

Norms, semi-normsP |Y , 294| | , 20| |bv , 286, 378, 554

‖ ‖ , 36‖ ‖∞ , 372, 523‖ ‖µ , 476‖ ‖1 , 6, 283‖ ‖∞ , 6, 249, 282, 283‖ ‖µ , 476‖ ‖i , 483

‖ ‖Ai , 483

‖ ‖p , 282, 283‖ ‖X′′ , 288‖ ‖bs , 6‖ ‖bv , 6, 286h(A) , 51qj , 349

Sequence spacesC , 16M(E, F ) , 394Ω, 16Πr , 339χ , 7χE , 557δ , 340` , 6, 283`(µ) , 476`1 , 6, 283`∞ , 5, 270`p , 283κ , 339FA , 537T , 7, 20, 40, 41, 47, 60, 93, 125,

149, 558ω , 5, 273bω , 228ϕ , 6bs , 6, 292bv , 6bv0 , 6c , 6c0 , 6c0µ , 476cs , 6, 292bcs , 229d , 339dr , 340f , 17, 19, 20, 55, 56, 69, 95, 359,

374, 558f0 , 20, 56, 61, 97, 463m , 5m0 , 7, 40, 41, 52, 94, 95, 121, 343,

368, 542, 548mµ , 476

Sequences/coefficients(fj) , 67

Index 577

Sαn (x) , 103

[A]nk , 13[x]k , 5∆nxk , 136σA

n (f, t) , 247e , 7en , 7s(f, t) , 247

sf (z) , 216

sfk(z) , 216

sn(f, t) , 245

x[n] , 318, 355Subsets/members of R, C

D1 , 10Dr , 217GS[f ] , 217Gα , 224K1 , 226S[f ] , 217Wϕ0 , 207[a] , 117K , 5N , 5N0 , 5N0

n , 28Nn , 134=a , 53<a , 53χ(A) , 46χ(f) , 379, 411δnk , 7S , 462C , 241ρ(T ) , 236σ(T ) , 236

Summability methods, matrices(H, pn) , 137(N, p) , 126(N, pn) , 126(R, p) , 112(R, pn) , 112bA , 228A1 , 10, 15, 21, 23, 24, 55, 158,

162, 190, 192, 194–196,210, 405, 406

Aα , 162At

n , 249B(α, β) , 162B0 , 15B∗

0 , 15B1 , 15, 24, 159, 160, 163, 165,

190, 197, 203, 405, 406

B∗1 , 14, 24, 49, 51, 59, 153, 222,

248Bα , 162, 196, 197Bp , 159, 160C1 , 8, 9, 13, 14, 20, 21, 23, 49, 51,

55, 58, 82, 86, 87, 91, 104,109, 112, 113, 116, 127,134, 168, 170–172, 178,194, 195, 248, 423, 444

Cα , 104–112, 126–128, 140, 148,149, 151, 162, 163, 172,175–177, 194, 196, 233,255, 256, 406, 479, 536

Eα , 140, 141, 147, 151, 153, 155,157, 163, 202, 203, 226,235, 406

F , 17Hα , 100, 101, 106, 111, 139, 148,

149, 151, 176Hp , 137–152, 223I , 14Jp , 158K[α, β] , 154L , 162Np , 126–135, 149, 152Pp , 158–165, 186–191Pp,tn , 159Rp , 112–126, 178–186, 558Sβ , 153Sf , 152Tα , 153Zα , 127Z 1

2, 8, 12, 14, 20, 21, 23, 39, 49,

51, 55, 424∆, 137Σ, 38Σ−1 , 38A , 19diag , 60

Topologiesβ(X, Y ) , 320η(X, Y ) , 351γ , 517σ(X, Y ) , 299, 315σγ(X, Y ) , 354τ(X, Y ) , 320τP , 293τω , 273, 294, 349

General Index

AB, 356, 357AB-space, 357

578 Index

Abel’s convergence test, 27Abel’s partial summation formula,

27Abel’s test, 28Abel’s theorem, 10Abel-type method, 158Abel method A1 , 15Abel method, generalized, 162Abschnittsdichte, 356absolutely µ-summable sequences,

476absolutely A-bounded domain, 81absolutely A-bounded sequences, 81absolutely p-summable sequence,

283, 304absolutely convex set, 283absolutely equivalent matrices, 482absolutely summable sequence, 6absolute sp ghp, 542, 543, 546absolute sp oscp, 472, 474, 542,

543absolute strong pointwise gliding

hump property, 542absolute strong pointwise oscillation

property, 472absolute summability, 25absorbing set, 283AD-space, 356adherent point, 265, 300adherent point of a net, 550adjoint map, 292AK, 355, 357AK-space, 357algebraic dual, 284almost convergent, 17almost coregular matrix, 421, 423α-dual, 341α-space, 342analytic

function, 340sequence, 339

analytic continuation, 214along a sequence of circles, 214

Antosik–Mikusinski matrix theo-rem, 562

application domain, 13, 229, 398associative matrix, 417associative part, 407asymptotic equal, 109

B0-invariant, 554B-invariant, 554b-comparison, 530

b-consistent summability methods,23

b-equivalent summability methods,22

b-stronger summability method, 22b-weaker summability method, 22Baire space, 279balanced set, 283Banach, theorem of, 333Banach–Steinhaus theorem, 290,

291, 333, 336, 337Banach space, 280barrel, 336barrelled space, 336Bernstein’s basis polynomials, 150β-dual, 39, 341β-space, 342bi-inverse matrix, 37binomial coefficients, 101BK-space, 359BK-topology, 359block sequence, 461block sequence, 1-, 542Borel

kernel, 248matrix B∗

1 , 14method B1 , 14, 15method, generalized, 162

Borel-type method, 159boundary behaviour of a power se-

ries, 206bounded

domain, 13linear operators, 285maps, set of, 270partial sums, 6sequence, 6, 279set, 279, 281, 289, 301set, norm, 289set, pointwise, 289set, uniformly, 289variation, 6

bounded consistency theorem, 81,528, 537

bs-norm, 6bv-norm, 6

C1-matrix, 14C1-method, 13Cα-matrix, 104Cα-method, 104canonical neighbourhood basis of

zero, 293Cartesian product, 264

Index 579

Cauchy’s limit theorem, 9Cauchy’s theorem for double series,

33Cauchy integral formulae, 214Cauchy product, 31Cauchy sequence, 275, 327Cesaro

matrix C1 , 14matrix Cα , 104method C1 , 13method Cα , 104, 162sections, 393

characteristic function, 95characteristic of a matrix, 46closed balls, 270closed graph, 331closed graph lemma, 332closed graph theorem, 332, 335, 337,

521closed graph theorem, Kalton’s, 335closed map, 331closed subset, 265closure of a set, 265cluster point, 68, 265, 300coarser topology, 267coercive method (matrix), 21cofinal subnet, 550cofinal subset, 550column-finite matrix, 82column condition, 44compact convergence, 215compactly summable series, 235compact set, 268, 551compact sets in K-spaces, 551compact space, 268compact space, sequentially, 279comparison of domains, 22compatible with the dual pair, 318complete

semi-metric space, 275semi-metrizable locally convex

space, 329semi-normed space, 280

completion of a normed space, 281conservative for null sequences, 21conservative method (matrix), 21consistency theorem, 464consistent summability methods, 23continuation, analytic, 214continuous, sequentially, 267continuous map, 266, 300continuous summability method,

158conull matrix, 49, 213

convergence factor sequence, 39convergent

net, 550sequence, 6, 264sequence, pointwise, 290series, 229

convergent double sequence, 16convex hull, 470convexity theorem, 176convex set, 283convolution, 129coordinatewise

convergence, 301product of sequences, 39sum, 461

coregular matrix, 49

decomposition of linear functionals,309

Dedekind’s test, 28definiteness, 269, 280dense subset, 93, 265diagonal matrix, 60diameter, 488difference matrix, 137differences of order k , 136direct sum, 6Dirichlet’s test, 28Dirichlet kernel, 246Dirichlet series, 34discrete

power series method, 159summability method, 158

discrete Borel method B∗1 , 14

discrete metric, 271discrete topology, 262disjointly supported, 94distance, 269distinguished subspace, 356, 386–

393distributive laws, 35domain (in C ), 214domain of a

matrix method, 13, 97, 397summability method, 12

double sequences, 16dual, 284, 307dual map, 292dual pair, 314dual space, 284, 307Du Bois–Reymond’s test, 28

Einfolgeverfahren, 73entire function, 341

580 Index

entire sequence, 340ε–δ-criterion, 274ε-neighbourhood, 270ε–n0-criterion, 273equicontinuous, 289, 519equicontinuous, pointwise, 289equicontinuous, uniformly, 289equicontinuous family, 312equiconvergent, 22, 91equipotent, 22equivalent semi-metric, 274equivalent semi-norms, 285equivalent summability methods, 22Euclidean

distance, 269metric, 269

Euclidean distance, 269Euler

matrix Eα , 140matrix, generalized, 233method Eα , 140, 163

Euler–Knopp method Eα , 140eventually constant sequence, 265

F-space, 329factor algebra of a regular matrix,

537factor sequence, 61factor sequence of a regular matrix,

537FAK, 355, 357FAK-space, 357family (of semi-norms), 293FC-effective matrix, 249Fejer kernel, 248Fejer’s theorem, 4, 255FF-form, 229FF-method, 228, 229finer topology, 267finitely non-zero sequences, 6finite subcovering, 268first category, subset of, 279fixed point form, 236FK-product, 394FK-space, 359, 361FK-topology, 359, 361Fourier-effective matrix, 249Fourier coefficients, 244Fourier series, 244Frechet combination, 272, 303Frechet space, 329functional sectional convergence,

355fundamental set, 313

γ-convergence, 517γ-dual, 341γ-space, 342gamma function, 105, 148gap sequence, 172

relative to p , 182gap Tauberian

condition, 172theorem for C1 , 172theorem for Rp , 182

generalizedAbel method, 162Borel method, 162Euler matrix, 233Euler method, 233

generated topology, 293generation of FK-spaces by linear

maps, 365gliding hump

method, 42property, 463property, absolute strong, 542property, signed weak, 562property, weak, 562

graph, 331

Hahn–Banach extension principle,287

Hahn–Banach theorem, 287, 308Hahn property, 558Hahn property, matrix, 94Hα-matrix, 100Hα-method, 100Hausdorff

matrix Hp , 137method Hp , 137methods, potent, 149

Hausdorff space, 268Holder

matrix Hα , 100, 139method Hα , 100, 139

hump, 42hyperplane, 312

i-semi-norm, 483identity matrix, 14IFK-space, 367inclusion map, 266, 268inclusion theorem, 21, 539, 552index sequence, 40indiscrete semi-metric, 271induced semi-metric, 271induced topology, 262inset, 78, 407

Index 581

internal inset, 416intersection method, 25intersection of FK-spaces, 362invariance of

PA , 453µ-uniqueness, 404, 453

invariant statement, 452inverse matrix, 37isometric isomorphism, 291

K-space, 350K-topology, 350Kalton’s closed graph theorem, 335Karamata’s Tauberian theorem for

Laplace transforms, 204Karamata matrix K[α, β] , 154kernel, 286

corresponding to a matrix, 247kernel,

Borel, 248Dirichlet, 246Fejer, 248

KG, 93Knopp, theorem of, 109Knopp and Schnee, theorem of, 111Kothe space, 342Kolmogoroff, theorem of, 302KP, 93Kronecker symbol, 90

`1-norm, 6Landau order symbols, 168Laurent matrix Sβ , 153lcs, 293Lebesgue constant, 250left inverse matrix, 37LFK-space, 367, 393limit formula, 44, 46, 51, 55, 80, 114,

127, 371, 373limit of

a net, 550a sequence, 265

limit superior, 51linear

form, 284functional, 284map, 284operator, 284

locally convexspace, 293space, sequentially complete, 329topology, 293

logarithmic method, 162logarithmic weighted mean, 116

lower triangular matrix, 37`p-norm, 283, 304

Mackeyspace, 320, 336, 521topology, 320

Mackey’s theorem, 322Mackey–Arens, theorem of, 322major rearrangement theorem, 32map, continuous, 266map, linear, 284mapping theorem, 394matrix, 13

Hahn property, 94, 557map, 13, 361method, 13method in RR-form, 229theorem, 562theorem, Antosik–Mikusinski,

562matrix, column-finite, 82maximal inset, 417Mazur–Orlicz, bounded consistency

theorem, 81Mazur–Orlicz type, theorem of, 462,

463, 540meagre subset, 279mean value

condition, 87property, 87, 109, 128, 418

Mercer’s theorem, 91metric, 269metric, discrete, 271metric, generated by a norm, 280metrizable locally convex space , 303Minkowski functional, 283, 295Mittag-Leffler star, 217mixed topology, 460, 517moment sequence, 144monotone sequence space, 542monotonicity of FK-topologies, 360µ-bounded domain, 478µ-bounded sequence, 476µ-comparison, 478µ-consistency, 478µ-consistent matrices, 478µ-continuous matrix, 455µ-equivalent matrices, 478µ-space, 455µ-stronger, 478µ-stronger matrix, 478µ-unique matrix, 421, 447, 448, 451,

453, 454µ-weaker matrix, 478

582 Index

multiplicative matrix, 419multiplier, 394, 537

space, 394

neighbourhood, 262basis, 263, 264collection, 263system, 264system of zero, 294

neighbourhoodsof zero, 294

net, 300, 550Neumann series, 237Norlund

matrix Np , 126method Np , 126

norm, 280normable metric space, 281normal matrix, 37normal topology, 351norm bounded set, 289normed space, 280nowhere dense, 279null domain, 13null sequence, 6

o–Tauberian theoremfor power series methods, 188for the Abel method, 190for the Borel method, 190

O–Tauberian theoremfor Riesz methods, 180for the Abel method, 192for the Borel method, 197for the Euler methods, 202

O–O-theorem for power series meth-ods, 190

O(1)–A1→C1-theorem, 194Okada’s theorem, 2201-block sequence, 542one-sequence method, 73one-sided oscillation Tauberian the-

orem for Rp , 178, 185open ball, 270open covering, 268open map, 331open mapping theorem, 331open set, 262operator

norm, 286semi-norm, 286

operator, linear, 2840R–Tauberian theorem for the Abel

method, 192

OR(1)–A1→C1-theorem, 195oscillation property, 472

P-perfect matrix, 444para-norm, 305para-norm, total, 305partial summation, 27perfect matrix, 78, 444perfect sequence space, 342ϕ-topology, 368PMI, 417pointwise

bounded family, 333bounded set, 289Cauchy sequence, 290convergence, 333convergent sequence, 290equicontinuous, 289limit, 290

polar topology, 319positive homogeneity, 280positive Norlund method, 129potent

Hausdorff methods, 149matrix (method), 93, 111, 117,

149, 557Riesz methods, 117

power series method, 158power series method Pp , 158power series method, discrete, 159power series methods, O–O-theorem

for, 190prime number theorem, 256product

metric, 272semi-metric, 272semi-norm, 281space, 264, 272topology, 264, 296

product ofa matrix and a sequence, 34FK-spaces, 394locally convex spaces, 296matrices, 35semi-normed spaces, 281series, 31

projection, 267, 296map, 349

quotient representation, 448, 459,484, 499, 531

quotient space, 282

R–F-equivalent matrices, 230

Index 583

rearrangement, 28regular for null sequences, 23regular summability method, 23relatively compact, 268relative topology, 262replaceable matrix, 412, 420–422,

454Riemann–Stieltjes integral, 144Riesz

matrix Rp , 112method Rp , 112methods, potent, 117

right inverse matrix, 37row-finite matrix, 36row norm condition, 44row sum condition, 46RR-form, 229RR-method, 228, 229RR-regular summation matrix, 230

SAK, 355, 357SAK-space, 357Saks space, 460, 516–522, 524, 526,

529scalar product of sequences, 34scaling, 94Schauder basis, 521Schur theorem, 51, 326, 479second category, subset of, 279second dual, 288sectional

boundedness, 92, 356convergence, 355

section density, 356section of a sequence, 355semi-definiteness, 269, 280semi-metric, 268

space, 269space, complete, 275

semi-metric, generated by a semi-norm, 280

semi-metric, indiscrete, 271semi-metrizable locally convex

space, 303semi-norm, 280semi-normable

locally convex space, 302semi-metric space, 281

semi-normedspace, 280space, complete, 280spaces, product of, 281

separable space, 288, 334

separating family of semi-norms,295

separationof sets, 312of sets, strict, 312

separation theorem, 312, 493sequences of zeros and ones, 7sequence space, 5sequence with bounded variation, 6sequential continuity, 274sequential dual, 356sequentially complete, 329, 521, 526,

528sequentially continuous, 267series, 228set of all double sequences, 16set of all sequences, 5signed p 01–oscp, 542, 543, 546,

557signed p ghp, 463, 464, 474, 475,

477, 542, 546signed p oscp, 462, 463, 472, 474,

477, 540–542signed pointwise

01-oscillating property, 542gliding hump property, 463oscillating property, 462

signed weak gliding hump property,562

simultaneouslyµ-consistent matrices, 500b-consistent matrices, 500consistent matrices, 500

singularity S1 , 504singularity S2 , 511singularity S3 , 513singularity S4 , 513slowly decreasing sequence, 169slowly increasing sequence, 169slowly oscillating relative to (Pn) ,

179slowly oscillating sequence, 62, 169SM-method, 19solid hull, 560solid sequence space, 342somewhere dense, 279Sonnenschein

matrix Sf , 152method Sf , 152

space, barrelled, 336spectral radius, 236spectrum, 236star, 217star-like set, 217

584 Index

statistical convergence, 25, 537step 1-block sequence, 462

with respect to an index sequence,461

Stirling’s formula, 59Stolz angle, 207Stone–Cech compactification, 537strong

subsequence, 544summability, 25topology, 320

strongersemi-metric, 274summability method, 22topology, 267

stronglyconservative method (matrix), 21,

55, 59, 94, 149, 374, 549,558

regular method (matrix), 55, 58,59, 77, 116, 125, 149–151

strongly A-summable, 353subset of first category, 279subset of second category, 279subspace

of a (semi-)normed space, 280of a locally convex space, 294of an FK-space, closed, 361of a semi-metric space, 271of a topological space, 262

substar, 217summability

function, 177functional, 12method, 12order, 86, 113

summablematrix series, 237sequence, 6, 12series, 12, 229, 235

summation domain, 229summation matrix, 38summation method, 229sum of FK-spaces, 362supremum metric, 270supremum norm, 6, 282symmetry, 269

T -statistical convergence, 25, 537Tauberian condition, 22, 168, 169Tauberian condition,

gap, 172local, 171one-sided local, 171

Tauberian theorem, 22for Laplace transforms, 204

Tauberian theorem, gap, 172Tauberian theorem for

A with MK(A), one-sided oscil-lation, 183

C1, one-sided oscillation, 170C1, oscillation, 170Rp, gap, 182power series methods, o–, 188the Abel method, OR–, 192the Abel method, o–, 190the Abel method, O–, 192the Borel method, o–, 190the Borel method, O–, 197, 202

Taylor matrix Tα , 153test function, 414theorem of

Abel and Stolz, 207Banach–Steinhaus type, 520Fejer, 4, 255Landau, 209Mazur–Orlicz type, 459, 463, 484,

529Mercer, 91

theorem on the inverse operator, 331thin sequence, 7Toeplitz

matrix, 393sections, 393

Toeplitz, Silverman, Kojima andSchur, theorem of, 46

Toeplitz–Silverman-type theorems,98, 368–375, 546–549

topologicaldual, 341isomorphism, 311product, 264, 296space, 262subspace, 262

topologically isomorphic, 311topology, 262topology, strong, 320topology generated by

a semi-metric, 270a semi-norm, 280

topology ofcoordinatewise convergence , 301the dual pair, 318uniform convergence on equicon-

tinuous subsets of X ′ , 320uniform convergence on the ele-

ments of M , 319total family of semi-norms, 295

Index 585

totally monotone sequence, 142transition matrix, 75translation, 94, 293translation invariance, 281triangle, 37triangle inequality, 269, 280triangle inequality, second, 269triangular matrix, lower, 37two-norm convergence, 517two-norm space, 516, 517type M, matrix of, 82–88, 101, 109,

113, 128, 141, 445, 447,448, 536

ultimately constant sequence, 265,339

uniform boundedness principle, 290uniformly bounded set, 289uniformly equicontinuous, 289union method, 25union of FK-spaces, 365–367, 393,

406, 556uniqueness of FK-topologies, 360unit vector, 7

V-summable, 12very conull matrix, 423, 433, 434,

444, 454

weak γ-dual topology, 354weaker

semi-metric, 274summability method, 22topology, 267

weak gliding hump property, 562weakly bounded subset, 319weak sectional convergence, 355weak topology, 299, 315, 316weighted mean, logarithmic, 116weighted mean method, 112wghp, 562Wiener’s Tauberian theorem, 204,

256Wiener theory, 204Wilansky property, 393, 561

zero matrix, 14ζ-dual, 341ζ-space, 342Zweier

matrix Zα , 127matrix Z 1

2, 14, 424, 541

method Zα , 127

method Z 12

, 13

List of Names

Abel, N. H., 4, 207Agnew, R. P., 67, 77, 78, 92, 140Alexiewicz, A., 515, 516, 522Andersen, A. F., 109, 176Antosik, P., 562Atalla, R. E., 537Bajsanski, B. M., 154Baker, J. W., 513Balser, W., 257Banach, S., 84, 338Baumann, H., 484Beekmann, W., ix, 25, 157, 204, 214,

452–454Benholz, M., 367, 394Bennett, G., 72, 94, 395, 439, 460,

522, 539, 561Bingham, N. H., 204Boos, J., 81, 367, 453, 455, 474, 484,

513, 529, 536, 562Borwein, D., 395Brauer, G., 537Brown, A., 166Brudno, A. L., 26, 78, 81, 522, 537Buntinas, M., 393–395Connor, J., 25, 395, 537Cooper, J. B., 516Copping, J., 72, 484, 498, 513Cramer, H., 256Darevsky, V., 73Eiermann, M., 243Fast, H., 25Faulstich, K., 226Fekete, M., 25Fleming, D. J., 81, 394, 474, 561, 562Ford, W. B., 126Fridy, J. A., 25Gao, X., 395Garling, D. J. H., 554Gawronski, W., 227Goes, G., 393–395Goldie, C. M., 204Große-Erdmann, K.-G., 227, 367,

394, 455Grothendieck, A., 518, 519Halmos, P. R., 166Hardy, G. H., 3, 25, 112, 144, 145,

166, 176, 191, 196Hausdorff, F., 101, 136, 140, 142,

144, 166

586 Index

Henriksen, M., 537Hille, E., 221Hurwitz, W. A., 140Jakimovski, A., 227, 395Kothe, G., 339, 341Kalton, N. J., 72, 334, 460, 522, 539Kangro, G. F., viii, 25Karamata, J., 154, 192, 204King, J. P., 227Kline, J., 25, 537Knopp, K., 109, 140Kojima, T., 46Kolk, E., 25Kratz, W., 188, 196Kublanowskaja, W. N., 228Kuttner, B., 41, 92, 94–96, 117, 149Landau, E., 175, 208Leibniz, G., 3Leibowitz, G., 166Leiger, T., 81, 393, 474, 562Le Roy, E., 226Littlewood, J. E., 25, 169, 176, 191,

196Lorch, L., 256Lorentz, G. G., 17, 19, 20, 51, 55,

136, 149, 177, 505, 513Luh, W., 226Luh, Y., 98Muller, J., 227Maddox, J., 25, 41, 92, 94, 95Magee, J. C., 394, 455Mahowald, M., 337Mazur, S., 26, 67, 69, 73, 78, 81, 338,

522Meyer-Konig, W., 203, 211, 212,

395, 439, 479Meyers, G., 393Mikusinski, J., 562Miller, H. I., 25Mittag-Leffler, G., 226Norlund, N. E., 126Neuser, R., 536Newman, D. J., 256Newton, Sir I., 3Niethammer, W., 227, 228, 242, 243Noll, D., 394, 561, 562Ohlenroth, W., 537Orhan, C., 25Orlicz, W., 26, 67, 69, 78, 81, 338,

460, 515, 516, 522Parameswaran, M. R., 92, 94, 96,

117, 149Paul, P. J., 561, 562Persson, A., 515, 516

Petersen, G. M., 19, 81, 484, 500,505, 512, 513, 537, 562

Peyerimhoff, A., 87Pringsheim, A., 25Ramanujan, M. S., 155Rhoades, B. E., 166Riesz, M., 112, 176, 211Ruckle, W. H., 368, 394, 455, 561Rudolf, O., 166Saxon, S. A., 394Schempp, W., 243Schur, I., 46, 51, 479Semadeni, Z., 515, 522Sharma, N. K., 166Shawyer, B., 25Shields, A. L., 166Silverman, L. L., 46, 136Siskakis, A. G., 166Sledd, W. T., 154, 156, 256Snyder, A. K., 395, 474, 561Sonnenschein, J., 100, 152Srinivasan, V. K., 256Stadler, W., 561Stadtmuller, K., 226Stadtmuller, U., 188, 197Stieglitz, M., 19, 98Stolz, O., 207Stoudt, G. S., 561Stuart, C. E., 562Swartz, C., 561, 562Tali, A., 536Tauber, A., 169Teugels, J. L., 204Tietz, H., 98, 192, 395Toeplitz, O., 339, 341Tomm, L., 226Trautner, R., 226, 227Varga, R. S., 243Watson, B., 25Wielandt, H., 192Wiener, N., 204Wilansky, A., 67, 72, 81, 144, 145,

259, 338, 386, 393, 395,396, 399, 420, 439, 454,455, 561

Wiweger, A., 515, 516Woronoj, G. Th., 126Zeller, K., ix, 25, 61–63, 67, 72, 73,

157, 192, 204, 214, 259,338, 355, 359, 393, 395,401, 425, 439, 453, 479,505, 513, 529, 537, 562

Zygmund, A., 255