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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