调和分析基础教程 第2版(英文影印版)

　　This book is intended as a primer in harmonic analysis at the upper undergraduate or early graduate level. All central concepts of harmonic analysis are introduced without too much technical overload. For example, the book is based entirely on the Riemann integral instead of the more demanding Lebesgue integral. Furthermore, all topological questions are dealt with purely in the context of metric spaces. It is quite surprising that this works. Indeed, it turns out that the central concepts of this beautiful and useful theory can be explained using very little technical background. ...
　　

i fourier analysis.
i fourier series
1.1 periodic functions
1.2 exponentials
1.3 the bessel inequality
1.4 convergence in the l2-norm
1.5 uniform convergence of fourier series
1.6 periodic functions revisited
1.7 exercises
2 hilbert spaces
2.1 pre-hilbert and hilbert spaces
2.2 l2-spaces
2.3 orthonormal bases and completion
2.4 fourier series revisited
2.5 exercises
3 the fourier transform
3.1 convergence theorems
3.2 convolution
3.3 the transform
3.4 the inversion formula

　　This book is intended as a primer in harmonic analysis at the upper undergraduate or early graduate level. All central concepts of harmonic analysis are introduced without too much technical overload. For example, the book is based entirely on the Riemann integral instead of the more demanding Lebesgue integral. Furthermore, all topological questions are dealt with purely in the context of metric spaces. It is quite surprising that this works. Indeed, it turns out that the central concepts of this beautiful and useful theory can be explained using very little technical background. .
　　The first aim of this book is to give a lean introduction to Fourier analysis, leading up to the Poisson summation formula. The second aim is to make the reader aware of the fact that both principal incarnations of Fourier theory, the Fourier series and the Fourier transform, are special cases of a more general theory arising in the context of locally compact abelian groups. The third goal of this book is to introduce the reader to the techniques used in harmonic analysis of noncommutative groups. These techniques are explained in the context of matrix groups as a principal example.
　　The first part of the book deals with Fourier analysis. Chapter 1 features a basic treatment of the theory of Fourier series, culminating in L2-completeness. In the second chapter this result is reformulated in terms of Hilbert spaces, the basic theory of which is presented there. Chapter 3 deals with the Fourier transform, centering on the inversion theorem and the Plancherel theorem, and combines the theory of the Fourier series and the Fourier transform in the most useful Poisson summation formula. Finally, distributions are introduced in chapter 4. Modern analysis is unthinkable without this concept that generalizes classical function spaces.
　　The second part of the book is devoted to the generalization of the concepts of Fourier analysis in the context of locally compact abelian groups, or LCA groups for short. In the introductory Chapter 5 the entire theory is developed in the elementary model case of a finite abelian group. The general setting is fixed in Chapter 6 by introducing the notion of LCA groups; a modest amount of topology enters at this stage. Chapter 7 deals with Pontryagin duality; the dual is shown to be an LCA group again, and the duality theorem is given. ..
　　The second part of the book concludes with Plancherel's theorem in Chapter 8. This theorem is a generalization of the completeness of the Fourier series, as well as of Plancherel's theorem for the real line. The third part of the book is intended to provide the reader with a first impression of the world of non-commutative harmonic analysis. Chapter 9 introduces methods that are used in the analysis of matrix groups, such as the theory of the exponential series and Lie algebras. These methods are then applied in Chapter 10 to arrive at a classification of the representations of the group SU(2). In Chapter, 11 we give the Peter-Weyl theorem, which generalizes the completeness of the Fourier series in the context of compact non-commutative groups and gives a decomposition of the regular representation as a direct sum of irreducibles. The theory of non-compact non-commutative groups is represented by the example of the Heisenberg group in Chapter 12. The regular representation in general decomposes as a direct integral rather than a direct sum. For the Heisenberg group this decomposition is given explicitly.Acknowledgements: I thank Robert Burckel and Alexander Schmidt for their most useful comments on this book. I also thank Moshe Adrian, Mark Pavey, Jose Carlos Santos, and Masamichi Takesaki for pointing out errors in the first edition. ...
　　Exeter, June 2004 Anton Deitmar