Harmonic and spectral analysis / László Székelyhidi.

Format:

Book

Author/Creator:

Székelyhidi, László, author.

Language:

English

Subjects (All):

Harmonic analysis.

Spectral sequences (Mathematics).

Physical Description:

1 online resource (247 p.)

Place of Publication:

Singapore : World Scientific, 2014.

Language Note:

English

Summary:

This book provides a modern introduction to harmonic analysis and synthesis on topological groups. It serves as a guide to the abstract theory of Fourier transformation. For the first time, it presents a detailed account of the theory of classical harmonic analysis together with the recent developments in spectral analysis and synthesis. Sample Chapter(s). Chapter 1: Duality of Finite Abelian Groups (254 KB). Contents: Abstract Harmonic Analysis: Duality of Finite Abelian Groups; Harmonic Analysis on Finite Abelian Groups; Set Theory and Topology; Invariant Means on Abelian Groups; Duality of

Contents:

Contents; Preface; Abstract Harmonic Analysis; 1. DUALITY OF FINITE ABELIAN GROUPS; 1.1 Characters; 1.2 Dual group; 2. HARMONIC ANALYSIS ON FINITE ABELIAN GROUPS; 2.1 Fourier transformation; 2.2 Convolution; 2.3 Convolution operators; 3. SET THEORY AND TOPOLOGY; 3.1 Basics from set theory; 3.2 Topological background; 3.3 Separation theorems, Uryshon's Lemma; 3.4 Compactification; 3.5 Partition of the unity; 3.6 Connectedness; 3.7 Topological groups; 3.8 Topological subgroups, factor groups; 3.9 Topological vector spaces; 3.10 The Minkowski functional; 3.11 Conjugate spaces

3.12 The Hahn-Banach Theorem3.13 The Stone-Weierstrass Theorem; 4. INVARIANT MEANS ON ABELIAN GROUPS; 4.1 Means on Abelian groups; 4.2 Invariant means; 5. DUALITY OF DISCRETE AND COMPACT ABELIAN GROUPS; 5.1 Characters on discrete Abelian groups; 5.2 Characters on compact Abelian groups; 5.3 Duality of discrete Abelian groups; 5.4 Convolution operators on compact Abelian groups; 5.5 Duality of compact Abelian groups; 6. DUALITY OF ELEMENTARY ABELIAN GROUPS; 6.1 The dual of some special Abelian groups; 6.2 Elementary Abelian groups; 7. HARMONIC ANALYSIS ON COMPACT ABELIAN GROUPS

7.1 The Riesz Representation Theorem7.2 Haar measure on the complex unit circle; 7.3 Fourier series; 7.4 Fourier analysis on compact elementary Abelian groups; 7.5 Fourier analysis on compact Abelian groups; 7.6 Integrable functions on compact Abelian groups; 7.7 Translation invariant spaces; 8. DUALITY OF LOCALLY COMPACT ABELIAN GROUPS; 8.1 Compactly generated locally compact Abelian groups; 8.2 The approximation theorem of compactly generated locally compact Abelian groups; 8.3 Duality theory of locally compact Abelian groups; 9. HAAR INTEGRAL ON LOCALLY COMPACT ABELIAN GROUPS

9.1 Haar measure and Haar integral9.2 The existence of Haar measure on locally compact Abelian groups; 9.3 The uniqueness of the Haar integral; 9.4 Convolution; 9.5 Haar measure on elementary and on compactly generated Abelian groups; 10. HARMONIC ANALYSIS ON LOCALLY COMPACT ABELIAN GROUPS; 10.1 Harmonic analysis on the group of integers; 10.2 Commutative algebras; 10.3 The Fourier transform of integrable functions; 10.4 Fourier transformation on locally compact Abelian groups; Spectral Analysis and Synthesis; 11. BASIC CONCEPTS; 11.1 Basics from ring theory; 11.2 Vector modules

11.3 Vector modules, group representations, and actions11.4 Spectral analysis and synthesis on vector modules; 11.5 Varieties on groups; 11.6 Annihilators; 12. BASIC FUNCTION CLASSES; 12.1 Exponentials; 12.2 Modified differences; 12.3 Automorphisms of themeasure algebra; 12.4 Generalized exponential monomials; 12.5 Generalized polynomials; 12.6 Generalized exponential polynomials; 12.7 Exponential monomials and polynomials; 12.8 Description of exponential polynomials; 12.9 An example; 13. THE TORSION FREE RANK; 13.1 Basics from group theory; 13.2 The torsion free rank and polynomials

13.3 The polynomial ring

Notes:

Description based upon print version of record.

Includes bibliographical references and index.

Description based on print version record.

ISBN:

981-4531-72-3