Székelyhidi Discrete Spectral Synthesis and Its Applications

1. Auflage 2007
ISBN: 978-1-4020-4637-7
Verlag: Springer-Verlag
Format: PDF
Kopierschutz: Adobe DRM (»Systemvoraussetzungen)

E-Book, Englisch, 119 Seiten

In order to study discrete Abelian groups with wide range applications, the use of classical functional equations, difference and differential equations, polynomial ideals, digital filtering and polynomial hypergroups is required. This book covers several different problems in this field and is unique in being the only comprehensive coverage of this topic. It should appeal to graduate students and researchers in harmonic analysis, spectral analysis, functional equations and hypergroups.

Autoren/Hrsg.

Székelyhidi, László

Inhaltsverzeichnis

1;Contents;7
2;Preface;9
3;1 Introduction;16
4;2 Spectral synthesis and spectral analysis;22
4.1;2.1 The basic problems of spectral analysis and spectral synthesis;22
4.2;2.2 Spectral analysis and synthesis on L1(G);23
4.3;2.2 Spectral analysis and synthesis on L (G);26
4.4;2.4 Spectral analysis and synthesis on C(G);32
5;3 Spectral analysis and spectral synthesis on discrete Abelian groups;40
5.1;3.1 Spectral analysis on discrete Abelian torsion groups;40
5.2;3.2 Spectral analysis on Abelian groups;42
5.3;3.3 Spectral analysis on commutative semigroups;42
5.4;3.4 Spectral synthesis and polynomial ideals;44
5.5;3.5 The failure of spectral synthesis on some types of discrete Abelian groups;49
5.6;3.6 Spectral synthesis on Abelian torsion groups;53
5.7;3.7 Polynomial functions and spectral synthesis;57
6;4 Spectral synthesis and functional equations;64
6.1;4.1 Convolution type functional equations;64
6.2;4.2 Mean value type functional equations;67
6.3;4.3 A functional equation in digital filtering;75
7;5 Mean periodic functions;84
7.1;5.1 The Fourier transform of mean periodic functions;84
7.2;5.2 The Fourier transform of exponential polynomials;92
7.3;5.3 Applications to differential equations;94
8;6 Difference equations in several variables;98
8.1;6.1 Spectral synthesis of difference equations;98
8.2;6.2 Applications;102
9;7 Spectral analysis and synthesis on polynomial hypergroups in a single variable;106
9.1;7.1 Polynomial hypergroups in one variable;106
9.2;7.2 Spectral analysis on polynomial hypergroups in one variable;111
9.3;7.3 Spectral synthesis on polynomial hypergroups in one variable;113
10;8 Spectral analysis and synthesis on multivariate polynomial hypergroups;118
10.1;8.1 Polynomial hypergroups in several variables;118
10.2;8.2 Exponential and additive functions on multivariate polynomial hypergroups;119
10.3;8.3 Spectral analysis and spectral synthesis on multivariate polynomial hypergroups;122
11;References;124
12;Index;128

Leseproben

(p. 1)

The basic tools for the investigation of different algebraic and analytical structures are representation and duality. "Representation" means that we establish a correspondence between our abstract structure and a similar, more particular one. Usually this more particular structure, the "representing" structure is formed by functions, de.ned on a set which is the so-called "dual" object.

In order to get a "faithful" representation, it seems to be reasonable that the correspondence in question is one-to-one. Another reasonable requirement is that if the same procedure is applied to the dual object, then its dual can be identified with the original structure. In order to do that, the dual object should have an "internal" characterization. Finally, a characterization of the "representing" structure is also desirable : which functions on the dual object belong to the "representing" structure?

The method of representation and duality appears in several different fields of algebra, analysis, etc. For instance, linear spaces can be represented as linear spaces of linear functionals, topological spaces can be represented as topological spaces of continuous functions, topological groups can be represented as topological groups of special homomorphisms, and so on. However, in all these cases one can assure the faithfulness via different assumptions only.

In the case of linear spaces the injectivity of the representing mapping holds only if the linear functionals of the original linear space form a separating family, which leads to Hahn– Banach type theorems. In the case of topological spaces the same requirement leads to conditions similar to those in Uryshon’s Lemma. In the theory of algebras the corresponding representation process can be described by the Gelfand transformation.

Let A be a complex algebra and let H denote a set of algebra homomorphisms of A onto C, the algebra of complex numbers. Such homomorphisms are called multiplicative linear functionals . We remark that the assumption on the surjectivity of a complex algebra homomorphism is obviously equivalent to it being nonidentically zero. Evidently, H is a subset of the algebraic dual of A, however, in general, H has no natural algebraic structure.