Harmonic analysis, abstract
The theory of abstract Fourier series and Fourier integrals (cf. Fourier integral). Classical harmonic analysis — the theory of Fourier series and Fourier integrals — underwent a rapid development, stimulated by physical problems, in the 18th century and 19th century; P. Dirichlet, B. Riemann, H. Lebesgue, M. Plancherel, L. Fejér, and F. Riesz formulated harmonic analysis as an independent mathematical discipline.
The further development of harmonic analysis resulted in the establishment of various relations between harmonic analysis and general problems in the theory of functions and functional analysis. The discovery of the Haar measure and the development of the theory of representations of infinite groups (cf. Representation of an infinite group), beginning with the work of H. Weyl and F. Peter  on the theory of representations of compact groups (cf. Representation of a compact group) and of L.S. Pontryagin  on the theory of characters of locally compact Abelian groups (cf. Character of a group), posed the problem of the natural limits of the main results of classical harmonic analysis. This problem is based on the following interpretation of an ordinary Fourier series in complex form. Let be a complex-valued square-summable function on a circle of unit length (or on the segment ), and let be its Fourier coefficients with respect to the system :
Then the Fourier series
of converges in the mean to in . The Lebesgue measure on generates the Haar measure on the circle (of length one), regarded as the group of rotations of the plane, while the functions represent a complete collection of irreducible unitary representations (cf. Unitary representation) of the topological group . For this reason all data involved in the definition of a Fourier series acquire a group-theoretic meaning, and it becomes possible to generalize the concept of a Fourier series on the basis of the theory of irreducible unitary representations of topological groups. Here, abstract harmonic analysis not only makes it possible to find a natural form for results of classical harmonic analysis on the real line or the circle, but also establishes new results regarding larger classes of topological groups.
Abstract harmonic analysis as the harmonic analysis on groups was developed mainly on the basis of the theory of characters of locally compact Abelian groups established by Pontryagin (, see also , , ). Abstract harmonic analysis is one of the natural fields of application of methods of the theory of Banach algebras, and may be regarded as being, to some extent, a branch of this theory. On the other hand, the framework of abstract harmonic analysis is a natural one for a number of classical problems in the theory of functions and functional analysis.
The applications of abstract harmonic analysis are extremely varied. The results are used in the general theory of locally compact groups (e.g. in structure theorems), in the theory of dynamical systems, in the theory of representations of infinite groups (which in its turn is one of the principal tools of abstract harmonic analysis), and in many other mathematical theories.
The best developed branch of abstract harmonic analysis is the theory of Fourier integrals on a locally compact Abelian group. A special type of non-commutative groups are the compact groups, the theory of representations of which is especially simple and complete: solutions of many classical problems of harmonic analysis have been obtained for compact groups. In the case of non-compact non-commutative groups the general theory is still far from complete (1989). However, even in this case one knows the natural limits of a number of fundamental results of classical harmonic analysis.
The connection between the problems of abstract harmonic analysis and the theory of Banach algebras is based on the fact that it is possible to construct two Banach algebras on each locally compact topological group , which both play a major role in the theory of representations of : the group algebra (cf. Group algebra of a locally compact group) and the measure algebra (cf. Algebra of measures), which is defined as follows. Let be the set of continuous functions on which vanish at infinity, and let be its dual space, the Banach space of bounded regular measures (cf. Regular measure) on . If a multiplication — the convolution — and an involution are introduced on by means of the relations
for all , then is converted into a Banach algebra with involution, which is called the measure algebra of the group . If is the left-invariant Haar measure on , the association to each element of the group algebra of the measure yields an isometric mapping of into a closed subalgebra of which preserves the involution. In this sense may be considered as a closed subalgebra of .
Abstract harmonic analysis on a locally compact Abelian group.
The following facts are required to construct a Fourier integral on a locally compact Abelian group . Any irreducible unitary representation of is one-dimensional and defines a continuous homomorphism from into the multiplicative group of complex numbers of modulus 1. Such a mapping is called a unitary character of . Let be the group of continuous characters of . Pontryagin's duality theorem states that the mapping
where , , is a topological isomorphism of onto , , , . The group is compact if and only if the group dual to it is discrete. The group of characters of the additive group of a non-discrete locally compact field is isomorphic to ; the group of characters of the group is isomorphic to the group of integers. If is a closed subgroup of and is the set of such that on , then is a closed subgroup of , , , and any unitary character of the subgroup can be extended to a unitary character of the group .
The Fourier integral on the group (or the Fourier transform on the group ) is the mapping under which a measure corresponds to the function on defined by the equation
The Fourier cotransform is the mapping defined by the equation
For the function is denoted by or (or, correspondingly, ). The mappings and are monomorphisms (cf. Monomorphism) of into ; the image of under these mappings is the algebra of linear combinations of continuous positive-definite functions on . The generalized Bochner theorem applies , : The function is a positive-definite function on if and only if is a positive measure, and then
where is the unit of .
The topological space is canonically homeomorphic to the spectrum of the ring (i.e. to the space of maximal ideals of the algebra ). In fact, with a character is associated the corresponding character of the commutative algebra defined by the formula
the Fourier cotransform is identical on with the Gel'fand representation of the Banach algebra . The spectrum of is usually not homeomorphic to .
Let be the Haar measure on and let be the corresponding Hilbert space. The following Plancherel theorem ,  is valid: If , then and, if the measures and are normalized in a certain way, then the mapping from the set into can be uniquely extended to a unitary operator from onto . This operator is known as the Fourier transform on . In such a case the measures and are called compatible. Let denote the linear subspace of generated by the functions of the form where . The following Fourier inversion formula ,  holds: If , then , and for all the equation
is valid, i.e. if is the canonical mapping of into , then for all . Let be the set of such that . Then the restriction of to is a one-to-one mapping of onto ; the inverse mapping is the restriction of to . If , then .
The classical Poisson summation formula is naturally interpreted in abstract harmonic analysis as follows. Let be a closed subgroup of the group . Let , , be the Haar measures on , and , respectively, normalized so that . Let be identified with and let be the Haar measure on compatible with . Finally, let and let the restriction of the continuous function to be integrable with respect to . Then the function on will be integrable with respect to the measure for almost-all , and
This formula is known as the generalized Poisson summation formula.
An important intrinsic problem in abstract harmonic analysis is the study of the Banach algebras and from the point of view of the Fourier transform on . The algebra is completely symmetric. The equality is valid if and only if is discrete. If is not discrete, contains non-symmetric maximal ideals. Let (respectively, ) be the set of Fourier transforms of elements of (respectively, ). and are function algebras on ; moreover, is a regular algebra, and if and only if for certain . The set of for which the support of the function is compact is a dense subset in .
The following results describe the functional properties of the Fourier transform on . Let be a function defined on , and let be non-discrete. Let act on , i.e. for any function with range in . will then be analytic on , and if is non-discrete, . Conversely, an analytic function on ( if is non-discrete) acts on . The function acts on if and only if is the restriction to of an entire real-analytic function. Let be defined on and let be an infinite discrete group. will act on if and only if and if is analytic in a certain neighbourhood of the origin (see ,  for a detailed list of references).
A traditional problem in the theory of Banach algebras is the structure and the properties of closed subalgebras. The following results concern closed subalgebras of the algebra . Let be a Borel semi-group in a locally compact Abelian group and let be the maximal subalgebra in . will then be contained in a closed semi-group inducing an Archimedean order on . A commutative Banach algebra is called a Stone–Weierstrass algebra if any one of its symmetric subalgebras separating the points of the spectrum of the ring and not vanishing simultaneously at any point of is dense in . is a Stone–Weierstrass algebra if and only if is totally disconnected.
One field of modern research in abstract harmonic analysis is the theory of thin sets (cf. Thin set) in locally compact Abelian groups, which may be regarded as a generalization of special results of classical harmonic analysis (in particular, the theory of lacunary trigonometric series). Let be a locally compact Abelian group and let be its unit element. A set is called independent if, for any and integers , either or . Any non-discrete locally compact Abelian group contains an independent set homeomorphic to a Cantor set. The independent sets include two important classes of sets, viz. Kronecker sets and -sets in a group. A set in a locally compact Abelian group is called a Kronecker set if for any continuous function on of modulus 1 and for any there exists a character such that
A Kronecker set is independent and contains no elements of finite order. Let be the cyclic group of order and let be the direct product of a countably-infinite number of groups isomorphic to . A set in is called a -set if any continuous function ( is considered to be a group of roots of unity) coincides on with some unitary character of the group . -sets are independent. If in each neighbourhood of the unit element of a locally compact group there is an element of infinite order, contains a Kronecker set which is homeomorphic to a Cantor set. If is a non-discrete locally compact Abelian group and if there exists a neighbourhood of the unit element without elements of infinite order, contains (for some ) as a closed subgroup; any group contains a -set which is homeomorphic to a Cantor set.
In finite-dimensional metrizable locally compact Abelian groups an independent set is a totally-disconnected set. An infinite-dimensional torus contains a Kronecker set homeomorphic to a segment. A union of two Kronecker sets on the circle may prove to be an independent set that is not a Kronecker set. By adding one point to some Kronecker set on an infinite-dimensional torus it is possible to obtain an independent set that is not a Kronecker set. If is a compact Kronecker set in and is a bounded measure concentrated on , then
Another important class of subsets of locally compact Abelian groups are Helson sets: Compact sets distinguished by the fact that every continuous function on is the restriction to of some element of the algebra . Any compact Kronecker set and any compact -set in are Helson sets. Not every compact subset of a locally compact Abelian group is a Helson set; there exist independent Cantor sets that are not Helson sets. A compact subset will be a Helson set if and only if and are equivalent norms on the Banach space of bounded measures on . Let denote the set of all for which for all . is then a closed ideal in . The space dual to is isometric to the space consisting of all for which
for any . A compact set is a Helson set if and only if any function is almost-everywhere equal to the Fourier transform of some bounded measure concentrated on . If is a Helson set in and if is a non-zero measure concentrated on , then does not tend to zero at infinity.
In the study of Fourier series on Abelian compact groups the concept of a Sidon set in discrete Abelian groups is very important. Let be a compact Abelian group and let be a subset of . A function is called an -function if for all . A linear combination of unitary characters on is called an -polynomial if is an -function. A set is called a Sidon set if there exists a constant such that
for any -polynomial on . The following assertions are equivalent:
a) is a Sidon set in ;
b) for any bounded -function the series is convergent;
c) for any continuous -function the series is convergent;
d) any bounded function on coincides with the restriction of some element to ;
e) any function on which tends to zero at infinity coincides with the restriction of some function to .
Any infinite set in contains an infinite Sidon set. Any independent subset in is a Sidon set.
Another field of abstract harmonic analysis, which at the time of writing is undergoing intensive development, is the theory of closed ideals in , in particular the theory of spectral synthesis. The problem of spectral synthesis may be posed in a general manner as follows. Let be a closed ideal in ; the problem is to clarify the conditions under which is the intersection of the maximal ideals in containing (it should be noted in this context that any maximal ideal in is regular, i.e. closed). One of the most important results of the theory of spectral synthesis is the Wiener Tauberian theorem: If is a closed ideal in , , then there exists a character such that for all . This theorem may be regarded as a positive solution of the problem stated above for the case . If every closed ideal in is the intersection of the maximal ideals in which it is contained, one says that satisfies spectral synthesis. A compact group satisfies spectral synthesis. On the other hand, the following theorem  is valid: If the group is non-discrete, does not satisfy spectral synthesis. It follows that if is non-discrete, then the algebra has non-symmetric closed ideals.
Abstract harmonic analysis on compact groups may be regarded as part of the theory of representations of compact groups; this theory is closely connected with the theory of almost-periodic functions on groups; see also Bohr compactification and the reviews in , . The problems of abstract harmonic analysis on an arbitrary locally compact topological group are much more complicated, in view of the insufficient development and complexity of the general theory of infinite-dimensional representations (cf. Infinite-dimensional representation) of a locally compact group. However, even in such a case the Fourier integral can be defined on a locally compact group  and it is possible to obtain analogues of the generalized Bochner theorem, the Plancherel formula and a number of other general theorems , .
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In the theory of thin sets an important type of problem is the question whether the union of two (or, occasionally, of countably many) sets of a certain type is again a set of that type. S.W. Drury proved [a1] that the union of two Sidon sets is again a Sidon set, and N.Th. Varopoulos [a2], using Drury's technique, proved the analogous result for Helson sets. For sets of spectral synthesis the problem is still (1989) not solved.
|[a1]||S.W. Drury, "Sur les ensembles de Sidon" C.R. Acad. Sci. Paris , A271 (1970) pp. 162–163|
|[a2]||N.Th. Varopoulos, "Sur la réunion de deux ensembles de Helson" C.R. Acad. Sci. Paris , A271 (1970) pp. 251–253|
|[a3]||W. Rudin, "Fourier analysis on groups" , Wiley (1962)|
|[a4]||H. Reiter, "Classical harmonic analysis and locally compact groups" , Clarendon Press (1968)|
|[a5]||L.-Å. Lindahl, F. Poulsen, "Thin sets in harmonic analysis" , M. Dekker (1971)|
|[a6]||C.C. Graham, O.C. McGehee, "Essays in commutative harmonic analysis" , Springer (1979) pp. Chapt. 5|
|[a7]||J. López, K. Ross, "Sidon sets" , M. Dekker (1975)|
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