## A First Course in Harmonic AnalysisThe 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 |

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### Contents

Fourier Series | 5 |

Hilbert Spaces | 25 |

The Fourier Transform | 41 |

Distributions | 59 |

Finite Abelian Groups | 73 |

LCA Groups | 81 |

The Dual Group | 101 |

Plancherel Theorem | 111 |

Matrix Groups | 129 |

The Representations of SU2 | 141 |

The PeterWeyl Theorem | 149 |

The Heisenberg Group | 157 |

A The Riemann Zeta Function | 175 |

187 | |

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### Common terms and phrases

assume bounded C+(G called Cauchy sequence Chapter character compact group compact support complete complex vector space compute continuous function continuously differentiable converges to f converges uniformly convolution countable define denote dense dual group example finite abelian group finite-dimensional follows Fourier coefficients Fourier series Fourier transform function f G CC(G given GLn(C group G group homomorphism Haar integral hence Hilbert space implies inner product interval 0,1 invariant integral irreducible unitary representation isometry isomorphic J—oo Jg Jg L2-norm LCA group Lemma Let f Let G Let H Lie algebra Lie(G linear map locally compact Matn(C metric space natural number norm oo oo orthonormal basis periodic function Plancherel theorem pre-Hilbert space Proof Proposition real numbers Riemann integrable satisfies sequence xn Show subgroup subset subspace tempered distribution tends to zero theory topology triangle inequality unique unitary representation