View source for Harmonic analysis, abstract

The theory of abstract [[Fourier series|Fourier series]] and Fourier integrals (cf. [[Fourier integral|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|Haar measure]] and the development of the theory of representations of infinite groups (cf. [[Representation of an infinite group|Representation of an infinite group]]), beginning with the work of H. Weyl and F. Peter [[#References|[1]]] on the theory of representations of compact groups (cf. [[Representation of a compact group|Representation of a compact group]]) and of L.S. Pontryagin [[#References|[2]]] on the theory of characters of locally compact Abelian groups (cf. [[Character of a group|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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h0464201.png" /> be a complex-valued square-summable function on a circle of unit length (or on the segment <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h0464202.png" />), and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h0464203.png" /> be its Fourier coefficients with respect to the system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h0464204.png" />:

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Then the Fourier series

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of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h0464207.png" /> converges in the mean to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h0464208.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h0464209.png" />. The Lebesgue measure on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h04642010.png" /> generates the Haar measure on the circle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h04642011.png" /> (of length one), regarded as the group of rotations of the plane, while the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h04642012.png" /> represent a complete collection of irreducible unitary representations (cf. [[Unitary representation|Unitary representation]]) of the topological group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h04642013.png" />. 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 ([[#References|[2]]], see also [[#References|[7]]], [[#References|[8]]], [[#References|[9]]]). 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h04642014.png" />, which both play a major role in the theory of representations of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h04642015.png" />: the group algebra (cf. [[Group algebra of a locally compact group|Group algebra of a locally compact group]]) and the measure algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h04642016.png" /> (cf. [[Algebra of measures|Algebra of measures]]), which is defined as follows. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h04642017.png" /> be the set of continuous functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h04642018.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h04642019.png" /> which vanish at infinity, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h04642020.png" /> be its dual space, the Banach space of bounded regular measures (cf. [[Regular measure|Regular measure]]) on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h04642021.png" />. If a multiplication — the convolution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h04642022.png" /> — and an involution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h04642023.png" /> are introduced on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h04642024.png" /> by means of the relations

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h04642025.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h04642026.png" /></td> </tr></table>

for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h04642027.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h04642028.png" /> is converted into a Banach algebra with involution, which is called the measure algebra of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h04642029.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h04642030.png" /> is the left-invariant Haar measure on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h04642031.png" />, the association to each element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h04642032.png" /> of the group algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h04642033.png" /> of the measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h04642034.png" /> yields an isometric mapping of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h04642035.png" /> into a closed subalgebra of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h04642036.png" /> which preserves the involution. In this sense <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h04642037.png" /> may be considered as a closed subalgebra of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h04642038.png" />.

==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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h04642039.png" />. Any irreducible unitary representation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h04642040.png" /> is one-dimensional and defines a continuous homomorphism from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h04642041.png" /> into the multiplicative group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h04642042.png" /> of complex numbers of modulus 1. Such a mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h04642043.png" /> is called a unitary character of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h04642044.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h04642045.png" /> be the group of continuous characters of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h04642046.png" />. Pontryagin's duality theorem states that the mapping

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where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h04642048.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h04642049.png" />, is a topological isomorphism of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h04642050.png" /> onto <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h04642051.png" /> [[#References|[2]]], [[#References|[3]]], [[#References|[4]]], [[#References|[6]]]. The group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h04642052.png" /> is compact if and only if the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h04642053.png" /> dual to it is discrete. The group of characters of the additive group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h04642054.png" /> of a non-discrete locally compact field is isomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h04642055.png" />; the group of characters of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h04642056.png" /> is isomorphic to the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h04642057.png" /> of integers. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h04642058.png" /> is a closed subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h04642059.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h04642060.png" /> is the set of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h04642061.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h04642062.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h04642063.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h04642064.png" /> is a closed subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h04642065.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h04642066.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h04642067.png" />, and any unitary character of the subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h04642068.png" /> can be extended to a unitary character of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h04642069.png" />.

The Fourier integral on the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h04642070.png" /> (or the Fourier transform on the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h04642071.png" />) is the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h04642072.png" /> under which a measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h04642073.png" /> corresponds to the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h04642074.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h04642075.png" /> defined by the equation

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h04642076.png" /></td> </tr></table>

The Fourier cotransform is the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h04642077.png" /> defined by the equation

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h04642078.png" /></td> </tr></table>

For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h04642079.png" /> the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h04642080.png" /> is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h04642081.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h04642082.png" /> (or, correspondingly, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h04642083.png" />). The mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h04642084.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h04642085.png" /> are monomorphisms (cf. [[Monomorphism|Monomorphism]]) of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h04642086.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h04642087.png" />; the image of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h04642088.png" /> under these mappings is the algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h04642089.png" /> of linear combinations of continuous positive-definite functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h04642090.png" />. The generalized Bochner theorem applies [[#References|[4]]], [[#References|[6]]]: The function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h04642091.png" /> is a positive-definite function on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h04642092.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h04642093.png" /> is a positive measure, and then

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h04642094.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h04642095.png" /> is the unit of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h04642096.png" />.

The topological space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h04642097.png" /> is canonically homeomorphic to the spectrum of the ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h04642098.png" /> (i.e. to the space of maximal ideals of the algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h04642099.png" />). In fact, with a character <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420100.png" /> is associated the corresponding character of the commutative algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420101.png" /> defined by the formula

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420102.png" /></td> </tr></table>

the Fourier cotransform <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420103.png" /> is identical on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420104.png" /> with the [[Gel'fand representation|Gel'fand representation]] of the Banach algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420105.png" />. The spectrum of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420106.png" /> is usually not homeomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420107.png" />.

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420108.png" /> be the Haar measure on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420109.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420110.png" /> be the corresponding Hilbert space. The following Plancherel theorem [[#References|[4]]], [[#References|[16]]] is valid: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420111.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420112.png" /> and, if the measures <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420113.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420114.png" /> are normalized in a certain way, then the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420115.png" /> from the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420116.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420117.png" /> can be uniquely extended to a unitary operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420118.png" /> from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420119.png" /> onto <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420120.png" />. This operator is known as the Fourier transform on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420122.png" />. In such a case the measures <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420123.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420124.png" /> are called compatible. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420125.png" /> denote the linear subspace of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420126.png" /> generated by the functions of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420127.png" /> where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420128.png" />. The following Fourier inversion formula [[#References|[4]]], [[#References|[16]]] holds: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420129.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420130.png" />, and for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420131.png" /> the equation

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420132.png" /></td> </tr></table>

is valid, i.e. if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420133.png" /> is the canonical mapping of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420134.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420135.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420136.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420137.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420138.png" /> be the set of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420139.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420140.png" />. Then the restriction of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420141.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420142.png" /> is a one-to-one mapping of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420143.png" /> onto <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420144.png" />; the inverse mapping is the restriction of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420145.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420146.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420147.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420148.png" />.

The classical [[Poisson summation formula|Poisson summation formula]] is naturally interpreted in abstract harmonic analysis as follows. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420149.png" /> be a closed subgroup of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420150.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420151.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420152.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420153.png" /> be the Haar measures on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420154.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420155.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420156.png" />, respectively, normalized so that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420157.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420158.png" /> be identified with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420159.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420160.png" /> be the Haar measure on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420161.png" /> compatible with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420162.png" />. Finally, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420163.png" /> and let the restriction of the continuous function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420164.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420165.png" /> be integrable with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420166.png" />. Then the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420167.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420168.png" /> will be integrable with respect to the measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420169.png" /> for almost-all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420170.png" />, and

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420171.png" /></td> </tr></table>

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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420172.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420173.png" /> from the point of view of the Fourier transform on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420174.png" />. The algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420175.png" /> is completely symmetric. The equality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420176.png" /> is valid if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420177.png" /> is discrete. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420178.png" /> is not discrete, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420179.png" /> contains non-symmetric maximal ideals. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420180.png" /> (respectively, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420181.png" />) be the set of Fourier transforms of elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420182.png" /> (respectively, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420183.png" />). <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420184.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420185.png" /> are function algebras on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420186.png" />; moreover, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420187.png" /> is a regular algebra, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420188.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420189.png" /> for certain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420190.png" />. The set of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420191.png" /> for which the support of the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420192.png" /> is compact is a dense subset in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420193.png" />.

The following results describe the functional properties of the Fourier transform on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420194.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420195.png" /> be a function defined on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420196.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420197.png" /> be non-discrete. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420198.png" /> act on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420199.png" />, i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420200.png" /> for any function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420201.png" /> with range in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420202.png" />. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420203.png" /> will then be analytic on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420204.png" />, and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420205.png" /> is non-discrete, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420206.png" />. Conversely, an analytic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420207.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420208.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420209.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420210.png" /> is non-discrete) acts on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420211.png" />. The function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420212.png" /> acts on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420213.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420214.png" /> is the restriction to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420215.png" /> of an entire real-analytic function. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420216.png" /> be defined on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420217.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420218.png" /> be an infinite discrete group. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420219.png" /> will act on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420220.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420221.png" /> and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420222.png" /> is analytic in a certain neighbourhood of the origin (see [[#References|[12]]], [[#References|[13]]] 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420223.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420224.png" /> be a Borel semi-group in a locally compact Abelian group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420225.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420226.png" /> be the maximal subalgebra in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420227.png" />. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420228.png" /> will then be contained in a closed semi-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420229.png" /> inducing an Archimedean order on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420230.png" />. A commutative Banach algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420231.png" /> is called a Stone–Weierstrass algebra if any one of its symmetric subalgebras <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420232.png" /> separating the points of the spectrum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420233.png" /> of the ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420234.png" /> and not vanishing simultaneously at any point of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420235.png" /> is dense in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420236.png" />. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420237.png" /> is a Stone–Weierstrass algebra if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420238.png" /> is totally disconnected.

One field of modern research in abstract harmonic analysis is the theory of thin sets (cf. [[Thin set|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|lacunary trigonometric series]]). Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420239.png" /> be a locally compact Abelian group and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420240.png" /> be its unit element. A set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420241.png" /> is called independent if, for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420242.png" /> and integers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420243.png" />, either <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420244.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420245.png" />. 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420249.png" />-sets in a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420250.png" /> group. A set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420251.png" /> in a locally compact Abelian group is called a Kronecker set if for any continuous function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420252.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420253.png" /> of modulus 1 and for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420254.png" /> there exists a character <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420255.png" /> such that

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420256.png" /></td> </tr></table>

A Kronecker set is independent and contains no elements of finite order. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420257.png" /> be the cyclic group of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420258.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420259.png" /> be the direct product of a countably-infinite number of groups isomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420260.png" />. A set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420261.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420262.png" /> is called a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420264.png" />-set if any continuous function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420265.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420266.png" /> is considered to be a group of roots of unity) coincides on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420267.png" /> with some unitary character of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420268.png" />. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420269.png" />-sets are independent. If in each neighbourhood of the unit element of a locally compact group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420270.png" /> there is an element of infinite order, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420271.png" /> contains a Kronecker set which is homeomorphic to a Cantor set. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420272.png" /> is a non-discrete locally compact Abelian group and if there exists a neighbourhood of the unit element without elements of infinite order, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420273.png" /> contains <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420274.png" /> (for some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420275.png" />) as a closed subgroup; any group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420276.png" /> contains a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420277.png" />-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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420278.png" /> is a compact Kronecker set in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420279.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420280.png" /> is a bounded measure concentrated on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420281.png" />, then

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420282.png" /></td> </tr></table>

Another important class of subsets of locally compact Abelian groups are Helson sets: Compact sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420283.png" /> distinguished by the fact that every continuous function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420284.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420285.png" /> is the restriction to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420286.png" /> of some element of the algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420287.png" />. Any compact Kronecker set and any compact <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420288.png" />-set in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420289.png" /> are Helson sets. Not every compact subset of a locally compact Abelian group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420290.png" /> is a Helson set; there exist independent Cantor sets that are not Helson sets. A compact subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420291.png" /> will be a Helson set if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420292.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420293.png" /> are equivalent norms on the Banach space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420294.png" /> of bounded measures on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420295.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420296.png" /> denote the set of all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420297.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420298.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420299.png" />. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420300.png" /> is then a closed ideal in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420301.png" />. The space dual to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420302.png" /> is isometric to the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420303.png" /> consisting of all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420304.png" /> for which

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420305.png" /></td> </tr></table>

for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420306.png" />. A compact set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420307.png" /> is a Helson set if and only if any function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420308.png" /> is almost-everywhere equal to the Fourier transform of some bounded measure concentrated on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420309.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420310.png" /> is a Helson set in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420311.png" /> and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420312.png" /> is a non-zero measure concentrated on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420313.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420314.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420315.png" /> be a compact Abelian group and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420316.png" /> be a subset of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420317.png" />. A function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420318.png" /> is called an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420320.png" />-function if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420321.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420322.png" />. A linear combination <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420323.png" /> of unitary characters on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420324.png" /> is called an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420326.png" />-polynomial if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420327.png" /> is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420328.png" />-function. A set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420329.png" /> is called a Sidon set if there exists a constant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420330.png" /> such that

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420331.png" /></td> </tr></table>

for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420332.png" />-polynomial on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420333.png" />. The following assertions are equivalent:

a) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420334.png" /> is a Sidon set in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420335.png" />;

b) for any bounded <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420336.png" />-function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420337.png" /> the series <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420338.png" /> is convergent;

c) for any continuous <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420339.png" />-function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420340.png" /> the series <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420341.png" /> is convergent;

d) any bounded function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420342.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420343.png" /> coincides with the restriction of some element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420344.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420345.png" />;

e) any function on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420346.png" /> which tends to zero at infinity coincides with the restriction of some function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420347.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420348.png" />.

Any infinite set in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420349.png" /> contains an infinite Sidon set. Any independent subset in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420350.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420351.png" />, in particular the theory of [[Spectral synthesis|spectral synthesis]]. The problem of spectral synthesis may be posed in a general manner as follows. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420352.png" /> be a closed ideal in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420353.png" />; the problem is to clarify the conditions under which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420354.png" /> is the intersection of the maximal ideals in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420355.png" /> containing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420356.png" /> (it should be noted in this context that any maximal ideal in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420357.png" /> is regular, i.e. closed). One of the most important results of the theory of spectral synthesis is the [[Wiener Tauberian theorem|Wiener Tauberian theorem]]: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420358.png" /> is a closed ideal in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420359.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420360.png" />, then there exists a character <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420361.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420362.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420363.png" />. This theorem may be regarded as a positive solution of the problem stated above for the case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420364.png" />. If every closed ideal in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420365.png" /> is the intersection of the maximal ideals in which it is contained, one says that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420366.png" /> satisfies spectral synthesis. A compact group satisfies spectral synthesis. On the other hand, the following theorem [[#References|[15]]] is valid: If the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420367.png" /> is non-discrete, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420368.png" /> does not satisfy spectral synthesis. It follows that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420369.png" /> is non-discrete, then the algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420370.png" /> 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|Bohr compactification]] and the reviews in [[#References|[11]]], [[#References|[4]]]. 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|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 [[#References|[5]]] and it is possible to obtain analogues of the generalized Bochner theorem, the Plancherel formula and a number of other general theorems [[#References|[8]]], [[#References|[11]]].

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  F. Peter,   H. Weyl,   "Die Vollständigkeit der primitiven Darstellungen einer geschlossener kontinuierlichen Gruppe"  ''Math. Ann.'' , '''97'''  (1927)  pp. 737–755</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  L.S. Pontryagin,   "The theory of commutative topological groups"  ''Ann. of Math. (2)'' , '''35''' :  2  (1934)  pp. 361–388  (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  E.R. van Kampen,   ''Proc. Nat. Acad. Sci. USA'' , '''20'''  (1934)  pp. 434–436</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  A. Weil,   "l'Intégration dans les groupes topologiques et ses applications" , Hermann  (1940)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  I.M. Gel'fand,   D.A. Raikov,   "Nondegenerate unitary representations of locally (bi)compact groups"  ''Mat. Sb.'' , '''13 (55)'''  (1943)  pp. 301–316  (In Russian)  (English abstract)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  D.A. Raikov,   "Harmonic analysis on commutative groups with the Haar measure and character theory"  ''Trudy Mat. Inst. Steklov.'' , '''14'''  (1945)  pp. 1–86  (In Russian)  (English abstract)</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  I.M. [I.M. Gel'fand] Gelfand,   D.A. [D.A. Raikov] Raikov,   G.E. [G.E. Shilov] Schilow,   "Kommutative Normierte Ringe" , Deutsch. Verlag Wissenschaft.  (1964)  (Translated from Russian)</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  M.A. Naimark,   "Normed rings" , Reidel  (1984)  (Translated from Russian)</TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top">  L.S. Pontryagin,   "Topological groups" , Princeton Univ. Press  (1958)  (Translated from Russian)</TD></TR><TR><TD valign="top">[10]</TD> <TD valign="top">  N. Bourbaki,   "Elements of mathematics. Spectral theories" , Addison-Wesley  (1977)  (Translated from French)</TD></TR><TR><TD valign="top">[11]</TD> <TD valign="top">  J. Dixmier,   "<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046420/h046420371.png" /> algebras" , North-Holland  (1977)  (Translated from French)</TD></TR><TR><TD valign="top">[12]</TD> <TD valign="top">  H., et al. Helson,   "The functions which operate on Fourier transforms"  ''Acta Math.'' , '''102'''  (1959)  pp. 135–157</TD></TR><TR><TD valign="top">[13]</TD> <TD valign="top">  E. Hewitt,   K.A. Ross,   "Abstract harmonic analysis" , '''1–2''' , Springer  (1963–1970)</TD></TR><TR><TD valign="top">[14]</TD> <TD valign="top">  L.H. Loomis,   "An introduction to abstract harmonic analysis" , v. Nostrand  (1953)</TD></TR><TR><TD valign="top">[15]</TD> <TD valign="top">  P. Malliavin,   "Impossibilité de la synthèse spectrale sur les groupes abéliens non compacts"  ''Publ. Math. IHES'' , '''2'''  (1959)  pp. 61–68</TD></TR><TR><TD valign="top">[16]</TD> <TD valign="top">  M.G. Krein,   "Sur une généralisation du théorème de Plancherel au cas des intégrales de Fourier sur les groupes topologiques commutatifs"  ''Dokl. Akad. Nauk SSSR'' , '''30'''  (1941)  pp. 484–488</TD></TR></table>



====Comments====
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 [[#References|[a1]]] that the union of two Sidon sets is again a Sidon set, and N.Th. Varopoulos [[#References|[a2]]], using Drury's technique, proved the analogous result for Helson sets. For sets of spectral synthesis the problem is still (1989) not solved.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  S.W. Drury,   "Sur les ensembles de Sidon"  ''C.R. Acad. Sci. Paris'' , '''A271'''  (1970)  pp. 162–163</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  N.Th. Varopoulos,   "Sur la réunion de deux ensembles de Helson"  ''C.R. Acad. Sci. Paris'' , '''A271'''  (1970)  pp. 251–253</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  W. Rudin,   "Fourier analysis on groups" , Wiley  (1962)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  H. Reiter,   "Classical harmonic analysis and locally compact groups" , Clarendon Press  (1968)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  L.-Å. Lindahl,   F. Poulsen,   "Thin sets in harmonic analysis" , M. Dekker  (1971)</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  C.C. Graham,   O.C. McGehee,   "Essays in commutative harmonic analysis" , Springer  (1979)  pp. Chapt. 5</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  J. López,   K. Ross,   "Sidon sets" , M. Dekker  (1975)</TD></TR></table>