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
1729 Leonhard Euler
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
Formulaes to determine coefficients in the series by the
integral of the function considered
Method for interpolation, published 1753
Trigonometric series of a function has been
presented for the first time in 1750 to 1751
Daniel Bernoulli
1753
Bernoulli, D., "Reflexions et eclaricissement
sur les nouvelles vibrations des corde",
Memories de l'Academie Royale des
Sciences et Belles Letters, Berlin, 1753.
The first series decomposition of
a signal is due to Daniel Bernoulli
who showed that
The most common movement of a string in
a musical instrument is composed of
the superposition of an infinite
number of harmonic vibrations.
Jean Le Rond d’Alambert
1754
Series in cosine functions of the reciprocal value
of the mutual distance of two planets
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
an =
1
π
f ( x) cos nxdx
∫
π
π
−
In the book by Godfrey Harold Hardy
Divergent Series, AMS Bookstore, 2000
Clairaut, A.-C., Hist. de l'Acad. des Sci.,
Paris, 1754,545, ff, publ. 1759
Joseph-Louis Lagrange
1759 Lagrange, J.L., Misc. Taur., 1, 1759 = Euvres 1, 110.
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
Euler Again
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.
1777
Another trigonometric series of a function - the method
equal to that used nowadays
Euler, L., Nova Acta Acad. Sci. Petrop., 11, 1793, 114132, publ 1798 = Opera (1), 16, Part 1, 333-355.
Marc Antoine Parseval des Chênes
Parseval theorem
5 April 1799
∑
x
f ( x) 2 = ∑ S f ( w) 2
w
The inner product of two
vectors/signals is the same as the ℓ2
inner product of their expansion
coefficients.
Energy of a signal f = Sum of squares of expansion coefficients Sf
Mémoire sur les séries et sur l'intégration complète
d'une équation aux differences partielle linéaires du
second ordre, à coefficiens constans
Particular case of the Plancharel formula in 1910
Squire and Royalist, arrested 1792
Parseval - original statement
π
∞
∞
∞
1  ∞
inu 
− mu  
− nu 
imu  
An an
∑
 ∑ An e  ∑ am e  +  ∑ An e  ∑ am e  du
∫
2π 0  n 0=
n 0
=
 m 0 =
 n 0=
 m 0

∞
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.
Predecessors
1729 Leonhard Euler interpolation
1747
Leonhard Euler movement of planets
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
Parseval
∑
x
f ( x) 2 = ∑ S f ( w) 2
w
Jean Baptiste Joseph Fourier
December 12, 1807
Presentation at the French Academy
Propagation of Heat in Solid Bodies
Rejected but encouraged for continuing the work.
Competition by the French Academy
1812
The Mathematical Theory of the Laws of
the Propagation of Heat and the Comparison of
the Results of this Theory with Exact Experiment
Continuation of the Work
Laplace, Lagrange, Legendre
Monge, Lacroix, Poisson
Criticized for the lack of mathematical rigor and
rejected for publication in the Memoirs of the Academy
Most criticized by Laplace, Poission, and Lagrange
Jean Baptiste Biot worked on heat conduction in
1802 and 1803, work known to Fourier
1811 revised paper
Biot
Poisson
Lagrange
Laplace
Monge
Finally Published
1816
1817
Nominated for the Academy, rejected
Fourier become a Member of the Academy
1822
Permanent Secretary of the Division for
Mathematical Sciences
Published the book Théorie Analytique de la Chaleur
Paris, Firmin Didot,
Analytical Theory of Heat
(255 x 202 mm), pp [iv] xxii 639, with two engraved plates
Profound study of nature is the most fertile source of
mathematical discoveries
Fourier
Contents
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 (continued)
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.
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
First Professorships
In 1794, Fourier was selected
among the 500 candidates for new
teachers at the Normal School
just established in Paris
Lagrange
1795
A professorship at the prestigious
École Polytéchnique in Paris first
as a superintendent
of lectures on fortification, and then
as a lecturer on analysis
Monge
Travel to Egypt and Related Studies
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
Fourier in Grenoble
1801, Fourier returned to France to the position of
prefect of Départment of Isére in Grenoble
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.
Description de l'Égypte, 21 vol. (1808–25)
Return to Paris
1814
After the end of Hundred Days,
during the Restoration, Fourier ran
into trouble for his political past
Director of the Bureau of Statistics
1831
Analyse des équations déterminées,
published by his friend Louis Marie Navier,
where he anticipated linear programming
Siméon Denis Poisson
1806
1808
Full Professor at the Ecole Polytechnique
in succession to Fourier who went to Grenoble.
Sur les inégalités des moyens mouvements des planètes
the mathematical problems which Laplace and Lagrange
had raised about perturbations of the planets.
Poisson followed an approach to these problems to use
series expansions to derive approximate solutions.
1813 - 1823
Foundations for the work by
Dirichlet and Riemann
Journal of the École Polytechnique
Memoirs de l’Academie
Poisson
Baron in1821, never either took out his diploma or used the title
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
Johann Peter Gustav Lejeune Dirichlet
1828
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
Founder of the theory of Fourier series
Georg Friedrich Bernhard Riemann
Correct mathematical formulation continued in the work of
Riemann
Work by Dirichlet characterized as
The first profound paper about this subject
Riemann, a student by Dirichlet, in his
habilitation thesis about representability
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
1902
Lebesgue integral
1904
Book by Lebesgue
in his PhD Thesis
Foundations for the formulation of
the Riesz-Fischer theorem in 1907
Riesz-Fischer Theorem
1907
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>.
Riesz
Fischer
Michel Plancherel
1910
DFT
Plancherel formula
If a function f is in both L1(R) and L2(R),
then its Fourier transform is in L2(R)
xk , y k
X j , Y j = F {xk , yk }
n −1
∑x
k =0
n −1
k
yk = ∑ X k Yk
j =0
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
Plancherel measures
1910
Privatdozent at the University of Geneva
Abstract Harmonic Analysis
Fourier analysis
Fourier series and Fourier transforms
Harmonic analysis
A generalization of Fourier series and Fourier transforms.
A branch of mathematics which studies the representation of
functions or signals as the superposition of basic waves. The
basic waves are called harmonics.
Abstract harmonic analysis
A generalization of harmonic analysis.
A branch of harmonic analysis that extends the definition of
the Fourier transforms to functions defined on various groups.
Julius Wihelm Richard Dedekind
This image cannot currently be displayed.
1870
Group Characters
Characters of a finite Abelian group
Group determinant = Determinant of the group matrix
G = ({g 0 ,..., g n −1},)
(n × n) Matrix
(i, j )-th element is xg0 g −j 1
{x
{x
g
g0
,..., xg gn−1
}
| g ∈ G} -n independent variables
over a field K
set of commuting variables
Ferdinand Georg Frobenius
April 12, 1896
Letter of Dedekind to Frobenius
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.
July 12, 1896
1897
Presentation of a paper on group characters
Berlin Academy
Group characters formulated by Frobenius
Fedor Eduardovich Molin
Theodor Molien
Notion of group ring in a study of group representations
Stuidied irreducible representations
Molien and Frobenius studied work of each other
Frobenius recommended work by Molien to Dedekind
in a letter of 24 February 1898
Riga
10 September 1861
Alfred Young
1900
Young diagrams and tables - a convenient way of
describing irreducible representations and for
manipulating with irreducible representations
The paper reviewed by Burnside
Used by Frobenius in 1903 in study of
representations of the symmetric group
1927
Young extended the work by Frobenius
1952
Further extensions
Hermann Weyl used Young diagrams in his book
Theory of Groups and Quantum Mechanics
Dover Publications, June 1, 1950
William Burnside and Issai Schur
1904-1905
Group Representations
Group representations = Column vectors
Linear transformations = Matrices
Burnside
1925
Schur Complete description of rational representations
of the general linear group GL(n,F)
F – a field, such as R or C
Schur
Student of Frobenius
Group of (n × n) invertible matrices over F
with the group operation as
the matrix multiplication
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
1880 Paper on transformation group
Lie groups = continuous transformation groups
Sophus Lie, Friedrich Engel
Theorie der Transformationsgruppen, three volumes, 1893
Groups
Finite
Abelian
Infinite
Non-Abelian
topological groups
Compact
Locally compact
Non-compact
Finite Abelian Groups
f ( x) = g −1 ∑ S f ( w) χ w ( x)
w∈Γ
Dual object
Γ = {χ w ( x)}, x ∈ G
the set of group characters of G
Abelian group under multiplication isomorphic to G
S f ( w) = ∑ f ( x) χ w ( x) −1
x∈G
Finite Non-Abelian Groups
K −1
f ( x) = ∑ Tr ( S f ( w)R w ( x))
w=0
Tr (Q) trace of Q
Dual object
Γ = {R w ( x)}, x ∈ G
(rw × rw ) matrices
the set of unitary irreducible representation of G
Fourier coefficients are (rw × rw ) matrices
g −1
S f ( w) = rw g −1 ∑ f (u )R w (u −1 )
u =0
Finite groups are compact groups
g=G
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
Peter-Weyl Theory
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
1933 Alfred Haar Existence of a right invariant integral for
locally compact groups
∫ f ( xa)dx = ∫ f ( x)dx
G
G
for all x ∈ G
Peter-Weyl Theorem
Infinite dimensional representation and its
decomposition by means of spectral theory for
bounded operators on Hilbert space
f ( x) =
∑
R w ∈Γ
rw −1
rw ∑ S (f i , j ) ( w) Rw( i , j ) ( x)
i , j =0
S (f i , j ) ( w) = ⟨ f , Rw( i , j ) ⟩ = ∫ f ( x)( Rw( i , j ) ) −1 ( x)dx
G
Plancharel Formula on Compact Groups
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Fourier series applies to square-integrable functions


2
f = ∫ f ( x) dx 
G

The norm
1
2
is finite
Fourier series for f(x) is equal to f in the mean-square sense of
f
2
=
rw −1
∑r ∑ S
Rw ∈Γ
w
i , j =0
(i , j ) 2
f
Plancharel formula = Criterion for extensions
Compact Abelian groups
All group representations are single-dimensional, i.e., reduce to
group characters
f ( x) = ∑ S f ( w) χ w ( x)
χ ∈Γ
S f ( w) = ∫ f ( x) χ w−1 ( x)dx
G
x+y = y + x
Hermann Klaus Hugo Weyl
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.
1923
Theory of compact groups in terms
of matrix representations
Compact Lie groups
fundamental character formula
1938
Locally Compact Abelian Groups
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.
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.
1
S f ( w) =
2π
∞
∫ f ( x )e
−iwx
dx
−∞
1
- Normalization of the Haar integral
2π
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.
If f ∈ L2, i.e., f is both integrable and square integrable, then Sf is also squareintegrable and f is equal to the Fourier integral in the means-square sense, i.e.,
the Plancharel formula is valid
∞
∫
−∞
2
S f ( w) dw =
∞
∫
−∞
2
f ( x) dx
Pontryagin Duality
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 Ĝ
When topologized with the topology of uniform
convergence of compact sets Ĝ is the dual group for G
Ĝ
ˆ
has also a dual group, called dual dual Gˆ
There is a canonical continuous homomorphism of G into Gˆˆ
If x ∈ G, then the corresponding member of
evaluated on a character
ˆˆ
G
χ w ∈ Gˆ has the value of χ w (x)
Lev Semenovich Pontryagin
1934
Exploited the structure theory and assumed that
the group is second countable and either
compact or discrete
1934
A member of Steklov Institute
1935
Head of the Department of Topology
and Functional Analysis
1939
A member of Academy of Science
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
Locally Compact Abelian Groups
G = locally compact Abelian group
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S f ( w) = ∫ f ( x) χ w ( x)dx
G
f ( x) = ∫ S f ( w) χ w ( x)dw
Gˆ
dx and dw suitably normalized
Haar integrals on G and
Ĝ
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
No general satisfactory theory currently known
The equivalent of Plancharel theorem
Infinite dimensional representations used
Many particular examples studied
The special linear group SL(n,F) is the subgroup of GL(n,F)
consisting of matrices with determinant 1.
Israil Moissevc Gelfand
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 representations of locally compact groups", Mat. Sb., Vol.
13, No. 55, 1943, 301-316
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.
Trigonometric Series - Predecessors
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1753
1729,1747,1753
Leonhard Euler
17.4.1707-18.9.1783
1754
Daniel Bernoulli
8.2.1700-17.3.1782
Jean Le Rond d'Alambert
16.11.1717-29.10.1783
∑
x
f ( x) = ∑ S f ( w)
2
w
∞
∑
n = −∞
cn
2
1
=
2π
π
∫π f ( x)
2
−
1799
1757
Alexis Claude Clairaut
7.5.1713 – 17.5.1765
2
Joseph-LouisLagrange
25.1.1736-10.4.1813
Marc-Antoine Parseval
27.4.1755-16.8.1836
sin x
cos x
sin 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
Fourier Analysis
Fourier series
∞
inx
e
c
∑n
f ( x) =
n = −∞
π
1
cn =
2π
∫π
f ( x)e −inx dx
−
Fourier transform
∞
f ( x) =
2πiwx
S
w
e
dw
(
)
f
∫
−∞
∞
S f ( w) =
∫
−∞
f ( x)e − 2πiwx dx
Abstract Harmonic Analysis
Johann Peter Gustav
Lejeune Dirichlet
13.2.1805-5.5.1859
Georg Friedrich
Bernhard Riemann
17.9.1826-20.7.1866
Simon Denis Poisson
Ernst Fischer
Frigyes Riesz
Henri Leon Lebesgue
21.6.1781-25.4.1840
28.6.1875-26.6.1941 22.1.1880 – 28.2.1956 12.6.1875 – 14.11.1954
Julius Wihelm Richard Dedekind
Ferdinand Georg Frobenius William Burnside
IssaiSchur
6.10.1831-12.2.1916
26.10.1849-3.8.1917 2.7.1852-21.8.1927 10.1.1875-10.1.1941
Michel Plancherel
16.1.1885-4.3.1967
Emmy Noether
23.3.1882-14.4.1935
Fourier Analysis
Group Representations
Hermann Weyl
Lev Semonovich Pontryagin Egbert Rudolf
9.11.1885-9.12.1955
van Kampen
3.9.1908-3.5.1988
28.5.1908-11.2.1942
AndréWeil
6.5.1906-6.8.1998
Compact non-Abelian
Locally Compact Abelian
Non-compact non-Abelian
Israil Moiseevic Gelfand
2.9.1913
Harish-Chandra
11.10.1923-16.10.1983
Closing Remarks
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
Anthony W. Knapp