A First Course in Harmonic Analysis

1. Front Matter

   Pages i-xii

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2. Fourier Analysis

   1. Fourier Series

      Pages 5-23

   2. Hilbert Spaces

      Pages 25-39

   3. The Fourier Transform

      Pages 41-57

   4. Distributions

      Pages 59-69

3. LCA Groups

   1. Finite Abelian Groups

      Pages 73-79

   2. LCA Groups

      Pages 81-100

   3. The Dual Group

      Pages 101-109

   4. Plancherel Theorem

      Pages 111-126

4. Noncommutative Groups

   1. Matrix Groups

      Pages 129-140

   2. The Representations of SU(2)

      Pages 141-147

   3. The Peter-Weyl Theorem

      Pages 149-156

   4. The Heisenberg Group

      Pages 157-173

5. Back Matter

   Pages 175-192

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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 ?rst 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 clas- ?cation 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 ?rst edition. Exeter, June 2004 Anton Deitmar LEITFADEN vii Leitfaden 1 2 3 5 4 6

• Fourier analysis
• Fourier transform
• Hilbert space
• Matrix
• Riemann integral
• analysis
• harmonic analysis
• set

• Department of Mathematics, University of Exeter, Exeter, Devon, UK

  Anton Deitmar

Professor Deitmar holds a Chair in Pure Mathematics at the University of Exeter, U.K. He is a former Heisenberg fellow and was awarded the main prize of the Japanese Association of Mathematical Sciences in 1998. In his leisure time he enjoys hiking in the mountains and practicing Aikido.

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