Remarks on History of Abstract Harmonic Analysis

Radomir S. Stanković, Jaakko T. Astola1, Mark G. Karpovsky2

Dept. of Computer Science, Faculty of Electronics, 18 000 Niš, Serbia

1Tampere International Center for Signal Processing Tampere University of Technology, Tampere, Finland

2Dept. of Electrical and Computer Engineering, Boston University

8 Saint Marry's Street, Boston Ma 02215, USA

Trigonometric Series Leonhard Euler 1729 Formulated and began to work on interpolation, the problem of determining function values in an arbitrary point x if its values for x = n, where n is an integer, are known.

Leonhard Euler

1747 Trigonometric series of a function derived from movement of planets, used method derived in 1729

Method for interpolation, published 1753

Trigonometric series of a function has been presented for the first time in 1750 to 1751

Formulaes to determine coefficients in the series by the integral of the function considered

Bernoulli, D., "Reflexions et eclaricissement sur les nouvelles vibrations des corde", Memories de l'Academie Royale des Sciences et Belles Letters, Berlin, 1753.

1753

The first series decomposition of a signal is due to Daniel Bernoulli who showed that

Daniel Bernoulli

The most common movement of a string in a musical instrument is composed of the superposition of an infinite number of harmonic vibrations.

1754 Series in cosine functions of the reciprocal value of the mutual distance of two planets

Jean Le Rond d’Alambert

d'Alambert, J. le R. "Researchers sur diferentes points importants du systeme du monde", 1754, Vol. 2, p. 66

Alexis Claude Clairaut

1757 Cosine series of a function derived in a study of the perturbations caused by the Sun

Clairaut, A.-C., Hist. de l'Acad. des Sci., Paris, 1754,545, ff, publ. 1759

1 ( )cosna f x nxdxπ

ππ −= ∫

In the book by Godfrey Harold Hardy Divergent Series, AMS Bookstore, 2000

Joseph-Louis Lagrange

Lagrange, J.L., Misc. Taur., 1, 1759 = Euvres 1, 110. 1759 From a letter by Lagrange to d'Alambert dated on August 15, 1768, it may be concluded that they considered representations of a non-periodic function

Lagrange, J.L., Euvres, 13, 116. Euvres, page 553, sine series Years 1762-1765 Edmund Taylor Whittaker, George Neville Watson, A Course of Modern Analysis: An Introduction to the General Theory of Infinite Processes and of Analytic Functions; with an Account of the Principal Transcendental Functions, Edition: 4 Published by Cambridge University Press, 1927

1777

Euler, L., Nova Acta Acad. Sci. Petrop., 11, 1793, 114- 132, publ 1798 = Opera (1), 16, Part 1, 333-355.

Another trigonometric series of a function - the method equal to that used nowadays

Euler, L., Nova Acta Acad. Sci., Petrop., 5, 1754-1755, 164-204, publ. 1760 = Opera (1), 14, 435-542-84

Euler, L., Opera (1), 15, 435-497.

Euler Again

Marc Antoine Parseval des Chênes 5 April 1799 Parseval theorem

Energy of a signal f = Sum of squares of expansion coefficients Sf

∑ ∑=x w

f wSxf 22 )()(

Particular case of the Plancharel formula in 1910

The inner product of two vectors/signals is the same as the ℓ2 inner product of their expansion coefficients.

Mémoire sur les séries et sur l'intégration complète d'une équation aux differences partielle linéaires du second ordre, à coefficiens constans

Squire and Royalist, arrested 1792

Parseval - original statement

0 0 0 0 00

12

inu mu nu imun n n m n m

n n m n mA a A e a e A e a e du

π

π

∞ ∞ ∞ ∞ ∞− −

= = = = =

= +

∑ ∑ ∑ ∑ ∑∫

Statement with no reference to a notion like Fourier series.

Entry on Parseval by H.C. Lennedy in C.S. Gillispie, (ed.), Dictionary of Scientific Biography, Vol. 10, Scribner’s Sons, New York, 1974, 327-328.

Leonhard Euler 1729

1747 movement of planets

interpolation

1753 Daniel Bernoulli movement of a string

1754 Jean Le Rond d’Alambert mutual distance of two planets

1757 Alexis Claude Clairaut movement of the Sun

1759 Joseph-Louis Lagrange

1777 Leonhard Euler

1799 Marc Antoine Parseval des Chênes ∑ ∑=x w

f wSxf 22 )()(

Leonhard Euler

Predecessors

Parseval

Jean Baptiste Joseph Fourier December 12, 1807 Presentation at the French Academy

Propagation of Heat in Solid Bodies

1812 Competition by the French Academy

The Mathematical Theory of the Laws of the Propagation of Heat and the Comparison of the Results of this Theory with Exact Experiment

Rejected but encouraged for continuing the work.

1811 revised paper

Criticized for the lack of mathematical rigor and rejected for publication in the Memoirs of the Academy

Continuation of the Work

Most criticized by Laplace, Poission, and Lagrange

Monge, Lacroix, Poisson Laplace, Lagrange, Legendre

Biot Poisson Lagrange Monge Laplace

Jean Baptiste Biot worked on heat conduction in 1802 and 1803, work known to Fourier

Finally Published

1817 1822 Permanent Secretary of the Division for

Mathematical Sciences

Published the book Théorie Analytique de la Chaleur

Fourier become a Member of the Academy

Profound study of nature is the most fertile source of mathematical discoveries

Fourier

Analytical Theory of Heat Paris, Firmin Didot, (255 x 202 mm), pp [iv] xxii 639, with two engraved plates

1816 Nominated for the Academy, rejected

I - Introduction. Exposition de l'objet de cet ouvrage. Notions générales et définitions préliminaires. Principe de la communication de la chaleur. Du mouvement uniforme et linéaire de la chaleur. Loi des températures permanentes dans un prisme d'une petite épaisseur. De l'échauffement des espaces clos. Du mouvement uniforme de la chaleur suivant les trois dimensions. Mesure du mouvement de la chaleur en un point donné d'une masse solide. II - Equation du mouvement de la chaleur. Equation du mouvement varié de la chaleur dans une armille; dans une sphère solide ; dans un cylindre solide. Equation du mouvement uniforme de la chaleur dans un prisme solide d'une longueur infinie. Equation du mouvement varié de la chaleur dans un cube solide. Equation générale de la propagation de la chaleur dans l'intérieur des solides. Equation générale relative à la surface. Application des équations générales. Remarques générales.

Contents

III - Propagation de la chaleur dans un solide rectangulaire infini. Exposition de la question. Premier exemple de l'usage des séries trigonométriques dans la théorie de la chaleur. Remarques sur ces séries. Solution générale. Expression finie du résultat de la solution. Développement d'une fonction arbitraire en séries trigonométriques. Application à la question actuelle. IV - Du mouvement linéaire et varié de la chaleur dans une armille. Solution générale de la question. De la communication de la chaleur entre des masses disjointes. V - De la propagation de la chaleur dans une sphère solide. Solution générale. Remarques diverses sur cette solution. VI - Du mouvement de la chaleur dans un cylindre solide. VII - Propagation de la chaleur dans un prisme rectangulaire. VIII - Du mouvement de la chaleur dans un cube solide. IX - De la diffusion de la chaleur. Du mouvement libre de la chaleur dans une ligne infinie ; dans un solide infini. Des plus hautes températures dans un solide infini. Comparaison des intégrales.

Contents (continued)

First Page

Fourier provided a solution of the problem considered by showing that the initial distribution of the temperature mast be expressed as a sum of infinitely many sine and cosine terms, which is now called the trigonometric or Fourier series.

Biography of Fourier

21.3. 1768 – 16.5.1830

Auxerre Paris

Military school run by Benedictines of Saint-Maur

French Revolution 1789-1799

Working as a publicist, recruiting agent, and a member of the Citizens Committee of Surveillance

Arrested in 1789 for defending victims of the terror of revolutionaries

In 1794, Fourier was selected among the 500 candidates for new teachers at the Normal School just established in Paris

A professorship at the prestigious École Polytéchnique in Paris first as a superintendent of lectures on fortification, and then as a lecturer on analysis

First Professorships

1795

Lagrange

Monge

In 1798, Monge and Fourier joined a group of cholars in the military campaign of Emperor Napoleon Bonaparte to Egypt

Fourier was appointed the governor of southern Egypt

Travel to Egypt and Related Studies

Appointed secretary of the Institut d‘ Egypts

In 1809 completed a major work on ancient Egypt, Préface historique.

In the same year, Napoleon awarded Fourier with the title of a Baron.

Fourier in Grenoble

Description de l'Égypte, 21 vol. (1808–25)

1801, Fourier returned to France to the position of prefect of Départment of Isére in Grenoble

1814 After the end of Hundred Days, during the Restoration, Fourier ran into trouble for his political past

Return to Paris

Director of the Bureau of Statistics

Analyse des équations déterminées, published by his friend Louis Marie Navier, where he anticipated linear programming

1831

1813 - 1823

Siméon Denis Poisson

Foundations for the work by Dirichlet and Riemann

Journal of the École Polytechnique

Memoirs de l’Academie Poisson

Sur les inégalités des moyens mouvements des planètes the mathematical problems which Laplace and Lagrange had raised about perturbations of the planets.

1808

Poisson followed an approach to these problems to use series expansions to derive approximate solutions.

Baron in1821, never either took out his diploma or used the title

1806 Full Professor at the Ecole Polytechnique in succession to Fourier who went to Grenoble.

Fourier and Poisson

Poisson has too much talent to apply it to the work of others. To use it to discover what is already known is to waste it ...

Fourier

Poisson was completely dedicated to mathematics.

Life is good for only two things, discovering mathematics and teaching mathematics.

Poisson

Dirichlet formulated conditions for a function f(x) to have the Fourier transform

f(x) must be single valued have a finite number of discontinuities in any given interval

have a finite number of extrema in any given interval

be square-integrable

1828

Johann Peter Gustav Lejeune Dirichlet

Founder of the theory of Fourier series

The first profound paper about this subject

Riemann, a student by Dirichlet, in his habilitation thesis about representability

Georg Friedrich Bernhard Riemann Correct mathematical formulation continued in the work of Riemann Work by Dirichlet characterized as

of functions by trigonometric series

If a function can be represented by a trigonometric series, what can one say about its behaviour

Conditions of a function to have an integral – Riemann integrability

Henri Leon Lebesgue

Lebesgue integral 1902 in his PhD Thesis

1904 Book by Lebesgue

Foundations for the formulation of the Riesz-Fischer theorem in 1907

Riesz-Fischer Theorem A function L2 is - (square-) integrable iff its Fourier series is L2-convergent. The application of this theorem requires use of the Lebesgue integral.

If {en} is an orthonormal basis for a real or complex infinite dimensional Hilbert space H, and {cn} a sequence of real or complex numbers such that Σ |cn|2 converges, then there is an x ∈H, such that x = Σ cnen and cn =<x,en>.

1907

Riesz Fischer

Michel Plancherel 1910 Plancherel formula

∑ ∑−

=

−

=

=1

0

1

0

n

k

n

jkkkk YXyxkk yx , { }kkjj yxFYX ,, =

Privatdozent at the University of Geneva 1910

Various orthonormal systems of functions, their summability and the representation of functions in such systems by Fourier series and Fourier integrals and more general integral transformations

DFT

Plancherel measures

If a function f is in both L1(R) and L2(R), then its Fourier transform is in L2(R)

A branch of harmonic analysis that extends the definition of the Fourier transforms to functions defined on various groups.

Abstract Harmonic Analysis

Fourier analysis

Harmonic analysis

A branch of mathematics which studies the representation of functions or signals as the superposition of basic waves. The basic waves are called harmonics.

Fourier series and Fourier transforms

A generalization of Fourier series and Fourier transforms.

Abstract harmonic analysis A generalization of harmonic analysis.

Group Characters 1870

Julius Wihelm Richard Dedekind

Characters of a finite Abelian group

Group determinant = Determinant of the group matrix

{ } ),,...,( 10 −= nggG )( nn× Matrix

),( ji -th element is 10 jg g

x −

{ }0 1,...,

gng gx x−

set of commuting variables

This image cannot currently be displayed.

{ }|gx g G∈ -n independent variables over a field K

July 12, 1896

Ferdinand Georg Frobenius

Presentation of a paper on group characters Berlin Academy

April 26, 1896 Letter of Frobenius to Dedekind

Irreducible characters for alternating groups A4, A5, symmetric groups S4, S5, and PSL(2,7) of order 168.

April 12, 1896 Letter of Dedekind to Frobenius

1897 Group characters formulated by Frobenius

Fedor Eduardovich Molin

Theodor Molien

Notion of group ring in a study of group representations

Molien and Frobenius studied work of each other Frobenius recommended work by Molien to Dedekind

Stuidied irreducible representations

Riga 10 September 1861

in a letter of 24 February 1898

Alfred Young Young diagrams and tables - a convenient way of describing irreducible representations and for manipulating with irreducible representations

Used by Frobenius in 1903 in study of representations of the symmetric group

1900

1927 Young extended the work by Frobenius

Hermann Weyl used Young diagrams in his book Theory of Groups and Quantum Mechanics

The paper reviewed by Burnside

1952 Further extensions

Dover Publications, June 1, 1950

Group Representations 1904-1905

William Burnside and Issai Schur

Group representations = Column vectors Linear transformations = Matrices

1925 Schur Complete description of rational representations of the general linear group GL(n,F)

F – a field, such as R or C

Group of (n × n) invertible matrices over F with the group operation as the matrix multiplication

Burnside

Schur Student of Frobenius

Emmy Noether

Matrices Linear transformations of a vector space

This approach is necessary for groups when infinite-dimensional representations are required as, for example, Lie groups

Sophus Lie

Lie groups = continuous transformation groups

Sophus Lie, Friedrich Engel Theorie der Transformationsgruppen, three volumes, 1893

1880 Paper on transformation group

topological groups

Non-compact Compact Locally compact

Groups

Finite Infinite

Abelian Non-Abelian

Finite Abelian Groups

∑Γ∈

−=w

wf xχwSgxf )()()( 1

{ } Gxxw ∈=Γ ,)(χ

the set of group characters of G

∑∈

−=Gx

wf xxfwS 1)()()( χ

Dual object

Abelian group under multiplication isomorphic to G

Finite Non-Abelian Groups

∑−

=

=1

0))()(()(

K

wwf xwSTrxf R )(QTr trace of Q

)( ww rr ×

∑−

=

−−=1

0

11 )()()(g

uwwf uufgrw RS

Fourier coefficients are matrices

Dual object

{ } Gxxw ∈=Γ ,)(Rthe set of unitary irreducible representation of G

)( ww rr × matrices

Gg =

Finite groups are compact groups

Compact Non-Abelian Groups

Peter, F., Weyl, H., “Die Vollstandigkeitder primitven Darstellungen einer geschlossen Kontinuirlichen Grouppe” Math. Ann., 9, 1927, 737-755.

Theory for compact groups that are not Lie groups still incomplete

Main contribution is the observation that not the finiteness of a group ensures existence of main properties of the Fourier representations, but existence of an averaging procedure over the group

Invariant integral that assigns a finite volume to the group

Peter-Weyl Theory

1933 Alfred Haar Existence of a right invariant integral for locally compact groups

∫ ∫=G G

dxxfdxxaf )()( for all x ∈ G

Peter-Weyl Theorem

Infinite dimensional representation and its decomposition by means of spectral theory for bounded operators on Hilbert space

∑ ∑Γ∈

−

=

=w

wr

ji

jiw

jifw xRwSrxf

R

1

0,

),(),( )()()(

∫ −=⟩⟨=G

jiw

jiw

jif dxxRxfRfwS )())((,)( 1),(),(),(

Plancharel Formula on Compact Groups

Fourier series applies to square-integrable functions

The norm is finite

This image cannot currently be displayed.

21

2)(

= ∫G

dxxff

Fourier series for f(x) is equal to f in the mean-square sense of

∑∑−

=Γ∈

=1

0,

2),(2 w

w

r

ji

jif

Rw Srf

Plancharel formula = Criterion for extensions

Compact Abelian groups

∑Γ∈

=χ

χ )()()( xwSxf wf

∫ −=G

wf dxxxfwS )()()( 1χ

All group representations are single-dimensional, i.e., reduce to group characters

x+y = y + x

Hermann Klaus Hugo Weyl

1923

Theory of compact groups in terms of matrix representations

Compact Lie groups fundamental character formula

1938

My work always tried to unite the truth with the beautiful, but when I had to choose one or the other, I usually chose the beautiful.

In topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space.

Locally Compact Abelian Groups

A topological space X is locally compact iff every point has a local base of compact neighborhoods

x+y = y + x

x+y = y + x x+y = y + x x+y = y + x

x+y = y + x x+y = y + x

Interpretation of Fourier Transform For f integrable over the real line, i.e., f ∈ L1, the spectrum Sf is well defined.

If f ∈ L2, i.e., f is both integrable and square integrable, then Sf is also square-integrable and f is equal to the Fourier integral in the means-square sense, i.e., the Plancharel formula is valid

However, the integrability of f does not imply the integrability of Sf , with integrability understood in the Lebesgue sense. Therefore, generalized methods of summability are required.

∫∫∞

∞−

∞

∞−

= dxxfdwwS f22

)()(

∫∞

∞−

−= dxexfwS iwxf )(

21)(π π2

1- Normalization of the Haar integral

For a locally compact Abelian group G, the set of unitary multiplicative characters under the pointwise multiplication expresses the structure of a locally compact Abelian group

G

GWhen topologized with the topology of uniform

is the dual group for G G

has also a dual group, called dual dual GThere is a canonical continuous homomorphism of G into

GIf x ∈ G, then the corresponding member of

Gwˆ∈χ has the value of )(xwχevaluated on a character

convergence of compact sets

G

Pontryagin Duality

Lev Semenovich Pontryagin

1934 Exploited the structure theory and assumed that the group is second countable and either compact or discrete

A member of Steklov Institute

Head of the Department of Topology and Functional Analysis

1939 A member of Academy of Science

1935

1934

Egbert Rudolf van Kampen 1935 Extensions of the work by Pontryagin

Pontryagin - Van Kampen duality

E.H. van Kampen, “On the connection between the fundamental groups of some related spaces”, American J. Math. 55, 1933, 261-267.

E.R van Kampen, E., "The structure of a compact connected group ", Amer. J. Math., 57, 1935, 301-308

This image cannot currently be displayed.G = locally compact Abelian group

∫=G

wf dxxxfwS )()()( χ

∫=G

wf dwxwSxfˆ

)()()( χ

dx and dw suitably normalized G and GHaar integrals on

Locally Compact Abelian Groups

Theory developed by Andre Weil

1938

André Weil

A famous anecdote, written in his autobiography says that when the Second World War started, he fled from France to Finland, however, was arrested there under suspicion of espionage, and was saved just by the intervention of Rolf Nevanlinna.

Non-Compact Non-Abelian Groups

Infinite dimensional representations used

No general satisfactory theory currently known

The equivalent of Plancharel theorem

The special linear group SL(n,F) is the subgroup of GL(n,F) consisting of matrices with determinant 1.

Many particular examples studied

Gelfand and Raikov in 1943, pointed out that, in principle, there should exists a sufficient number of irreducible representations to perform harmonic analysis on locally compact groups

Gelfand, I.M., Raikov, D.A., "Irreducible unitary rep- resentations of locally compact groups", Mat. Sb., Vol. 13, No. 55, 1943, 301-316

Israil Moissevc Gelfand

Harish-Chandra

Extensions of harmonic analysis to noncompact real semi-simple Lie groups

1952 Plancharel theorem

Harish-Chandra, "The Plancherel formula for complex semisimple Lie groups", Trans. Amer. Math. Soc., Vol. 76, No. 3, 1954, 458-528.

Leonhard Euler17.4.1707-18.9.1783

Jean Le Rond d'Alambert16.11.1717-29.10.1783

Daniel Bernoulli8.2.1700-17.3.1782

Joseph-LouisLagrange25.1.1736-10.4.1813

Alexis Claude Clairaut 7.5.1713 – 17.5.1765

1729,1747,1753

1753

1754

1757 1799

Marc-Antoine Parseval 27.4.1755-16.8.1836

This image cannot currently be displayed.

∑ ∑=x w

f wSxf22 )()(

Trigonometric Series - Predecessors

∑ ∫∞

−∞= −

=n

n xfcπ

ππ22 )(

21

sin x cos x sin x cos x sin x

cos x sin x

cos x sin x

cos x sin x

cos x sin x

cos x sin x

cos x sin x

cos x sin x

cos x sin x

Fourier Analysis

∑∞

−∞=

=n

inxnecxf )(

∫−

−=π

ππdxexfc inx

n )(21

Fourier series

Fourier transform

∫∞

∞−

= dwewSxf iwxf

π2)()(

∫∞

∞−

−= dxexfwS iwxf

π2)()(

Johann Peter Gustav Lejeune Dirichlet

13.2.1805-5.5.1859

Henri Leon Lebesgue 28.6.1875-26.6.1941

Georg Friedrich Bernhard Riemann

17.9.1826-20.7.1866

Simon Denis Poisson 21.6.1781-25.4.1840

Michel Plancherel 16.1.1885-4.3.1967

Julius Wihelm Richard Dedekind 6.10.1831-12.2.1916

Ferdinand Georg Frobenius 26.10.1849-3.8.1917

William Burnside 2.7.1852-21.8.1927

Emmy Noether 23.3.1882-14.4.1935

Issai Schur 10.1.1875-10.1.1941

Hermann Weyl 9.11.1885-9.12.1955

Lev Semonovich Pontryagin 3.9.1908-3.5.1988

Egbert Rudolf van Kampen

28.5.1908-11.2.1942

Harish-Chandra 11.10.1923-16.10.1983

Israil Moiseevic Gelfand 2.9.1913

Andr Weil 6.5.1906-6.8.1998

é

Abstract Harmonic Analysis

Fourier Analysis

Group Representations

Locally Compact Abelian

Compact non-Abelian

Non-compact non-Abelian

Frigyes Riesz 22.1.1880 – 28.2.1956

Ernst Fischer 12.6.1875 – 14.11.1954

Although harmonic analysis may be at the core of the solution of a problem, several layers of ingenious ideas may lie between the statement of the problem and the use of harmonic analysis

Closing Remarks

Anthony W. Knapp