In the book, the author gives a survey of discrete spectral analysis and discrete spectral synthesis. He presents the most recent results on this subject and their applications in different fields of mathematics with main emphasis on applications in the theory of functional equations.
In Chapter 1, the author shows the classical background of spectral analysis and synthesis in functional analysis. He presents elements of the Gelfand theory of commutative Banach algebras.
In Chapter 2, the basic problems of spectral analysis and synthesis are described. For commutative Banach algebras, they are formulated in the following way: is every proper closed ideal included in a regular maximal ideal (spectral analysis), is every proper closed ideal the intersection of all regular maximal ideals in which it is contained (spectral synthesis). For the algebra \(L^{\infty}(G)\) with the weak\(^{\ast}\)-topology (which is the dual of \(L^{1}(G)\)), where \(G\) is a locally compact Abelian group, the spectral analysis and synthesis problems have the following form: does every proper closed translation invariant subspace contain a character (spectral analysis), are the trigonometric polynomials of each proper closed translation invariant subspace dense in this subspace (spectral synthesis). This is a consequence of the fact that there exists a one-to-one correspondence between the closed ideals of \(L^{1}(G)\) and the closed translation invariant subspaces of \(L^{\infty}(G)\). The set of all characters in a closed translation invariant subspace \(V\) of \(L^{\infty}(G)\) is said to be the spectrum of \(V\). The spectrum of \(V\) is equal to the set of all common zeros of the Fourier transforms of the elements of \(L^{1}(G)\) which annihilate \(V\).
More general settings of the spectral analysis and synthesis problems are obtained replacing \(L^{\infty}(G)\) by a translation invariant topological vector space \(\mathcal{F}(G)\) of functions on a locally compact Abelian group \(G\) and \(L^{1}(G)\) by \(\mathcal{F}(G)^{\ast}\), the dual of \(\mathcal{F}(G)\). The author deals mostly with the case \(\mathcal{F}(G)=C(G)\), the space of all continuous complex valued functions on a locally compact Abelian group \(G\) equipped with the topology of uniform convergence on compact sets. Then \(\mathcal{F}(G)^{\ast}\) is the space \(\mathcal{M}_{c}(G)\) of all compactly supported complex Radon measures on \(G\). In this chapter, the author introduces the definitions of crucial notions used in spectral analysis and synthesis in \(C(G)\). A continuous homomorphism of \(G\) into the multiplicative group of nonzero complex numbers is called an exponential, and a continuous homomorphism of \(G\) into the additive group of complex numbers is called additive function. A function \(x\mapsto P(a_1(x),\dots,a_n(x))\) on \(G\), where \(P\) is a complex polynomial in \(n\) variables and \(a_1, \ldots,a_n\) are additive functions, is said to be a polynomial. A product of a polynomial and exponential is called an exponential monomial, and a linear combination of exponential monomials is called an exponential polynomial. For the space \(C(G)\), the spectral analysis and synthesis problems can be formulated in the following way: does every variety (i.e. a proper closed translation invariant subspace) in \(C(G)\) contain an exponential (spectral analysis), are the linear combinations of all exponential monomials of each variety dense in this variety (spectral synthesis). At the end of this chapter, the author presents classical results which say that spectral synthesis holds in \(C(\mathbb{R})\) and in \(C(G)\), where \(G\) is a finitely generated discrete Abelian group. The considerations included in this chapter show the connections of spectral analysis and synthesis to Tauberian theory.
In Chapter 3, the author deals with discrete spectral analysis and synthesis, i.e. spectral analysis and synthesis on discrete Abelian groups. This is the main part of the book. It contains most recent results on the considered subject. The first result presented here says that spectral analysis holds for discrete Abelian torsion groups \(G\) (i.e. spectral analysis holds for \(C(G)\), where \(G\) is a discrete Abelian torsion group). The next one is a characterization of Abelian groups having the spectral analysis property, namely spectral analysis holds for \(G\) if and only if the torsion free rank of \(G\) is less than the continuum. The first result of this chapter concerning spectral synthesis says that spectral synthesis does not hold on Abelian groups with infinite torsion free rank. Moreover, it is proved that spectral synthesis holds on every Abelian torsion group. On the basis of these facts, the author formulates the conjecture that spectral synthesis holds on an Abelian group if and only if its torsion free rank is finite. An equivalent form of this conjecture ends this chapter, namely it is proved that the torsion free rank of an Abelian group \(G\) is finite if and only if each complex bi-additive function \(B:G\times G \rightarrow \mathbb{C}\) is a bilinear function of complex additive functions, i.e. \[ B(x,y)=\sum_{i=1}^{n} \sum_{j=1}^{n} b_{ij} a_{i}(x) a_{j}(x), \] where \(n\) is a positive integer, \(b_{ij}\in \mathbb{C}\) and \(a_i\) is a complex additive function.
In Chapter 4, some applications of spectral synthesis in the theory of functional equations are discussed. The first application concerns systems of convolution type equations, i.e. the systems of equations \(f\ast\mu=0\) for all \(\mu \in \Lambda\), where \(\Lambda\) is a nonempty set of measures in \(\mathcal{M}_{c}(G)\) and \(f \in C(G)\) is an unknown function. Since the solution space of this system is a variety and for each variety in \(C(G)\) there exists a system of convolution type equations, the meaning of spectral synthesis for a given variety is that each solution of the corresponding system of convolution type equations can be uniformly approximated on compact sets by linear combinations of exponential monomial solutions of this system. Moreover, if spectral synthesis holds for \(C(G)\), then the problem of equivalence of two systems of convolution type equations can be reduced to the problem whether the spectral sets of the systems (i.e. the sets of exponential monomial solutions) are equal. Such a method is used in the next section of this chapter to show the equivalence of two mean-value type functional equations. In the last section of this chapter, an application is given of spectral synthesis to a functional equation in digital filtering.
In Chapter 5, the author deals with mean periodic functions, i.e. continuous functions \(f:G\rightarrow \mathbb{C}\), where \(G\) is a locally compact Abelian group, having the property that there exists a nonzero compactly supported complex Radon measure \(\mu\) on \(G\) such that \(f\ast\mu=0\). Based on the fact that spectral synthesis holds in \(C(\mathbb{R})\), a generalized Fourier transformation on the space of mean periodic functions is defined. This transformation is used to solve a convolution type functional equation on \(\mathbb{R}\). In the next section, an extension of this transformation to the set of all exponential polynomials on a locally compact Abelian group is formulated. This extension can be used to find exponential polynomial solutions of some ordinary and partial differential equations. Examples of this type are given in the last section of this chapter.
In Chapter 6, making use of the fact that spectral synthesis holds for \(G=\mathbb{Z}^{n}\), the author presents an application of spectral synthesis to systems of linear homogeneous difference equations with constant coefficients. In the language of difference equations, the spectral synthesis result means that the exponential monomial solutions of a system of linear difference equations generate a dense set in the solution space. Thus the spectral set of the system characterizes the whole solution space. The problem of determining the spectral set of the system can be reduced to that of finding the polynomial solutions of an appropriate system of linear homogeneous partial differential equations. Examples illustrating this method are given at the end of this chapter.
In Chapter 7, the author describes the notion of spectral analysis and synthesis on hypergroups. At the beginning of the chapter, we can find the definition of a hypergroup \(K\) and the definition of a variety in \(C(K)\). For an arbitrary commutative hypergroup \(K\), the problems of spectral analysis and synthesis are formulated in the following way: does each variety in \(C(K)\) contain a one-dimensional variety (spectral analysis), is each variety in \(C(K)\) the sum of finite-dimensional varieties (spectral synthesis). At the end of the first section of this chapter, a special type of hypergroups is defined, namely a polynomial hypergroup associated with a sequence of polynomials in one variable. In the next two sections, it is proved that spectral analysis and synthesis hold for polynomial hypergroups. In this chapter, one can also find the definition and some properties of exponential monomials and exponential polynomials on polynomial hypergroups.
In the last chapter, the notion of spectral analysis and synthesis for multivariate polynomial hypergroups (i.e. polynomial hypergroups associated with a family of polynomials in \(d\) variables) is presented. In the second section of this chapter, characterizations of exponential and additive functions on multivariate polynomial hypergroups are given. Then it is proved that spectral analysis and synthesis hold for multivariate polynomial hypergroups.
This book is an excellent source of information about spectral analysis and synthesis and their applications. It contains a long reference list, so the interested readers can extend their knowledge on this subject.