## Abstract Harmonic Analysis of Continuous Wavelet Transforms, Issue 1863This volume contains a systematic discussion of wavelet-type inversion formulae based on group representations, and their close connection to the Plancherel formula for locally compact groups. The connection is demonstrated by the discussion of a toy example, and then employed for two purposes: Mathematically, it serves as a powerful tool, yielding existence results and criteria for inversion formulae which generalize many of the known results. Moreover, the connection provides the starting point for a – reasonably self-contained – exposition of Plancherel theory. Therefore, the volume can also be read as a problem-driven introduction to the Plancherel formula. |

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### Contents

Introduction | 1 |

12 Overview of the Book | 4 |

13 Preliminaries | 5 |

Wavelet Transforms and Group Representations | 15 |

22 Coherent States and Resolutions of the Identity | 18 |

23 Continuous Wavelet Transforms and the Regular Representation | 21 |

24 Discrete Series Representations | 26 |

25 Selfadjoint Convolution Idempotents and Support Properties | 39 |

382 Construction Details | 99 |

Plancherel Inversion and Wavelet Transforms | 105 |

42 Plancherel Inversion | 113 |

43 Admissibility Criteria | 119 |

44 Admissibility Criteria and the Type I Condition | 129 |

45 Wigner Functions Associated to Nilpotent Lie Groups | 130 |

Admissible Vectors for Group Extensions | 139 |

51 Quasiregular Representations and the Dual Orbit Space | 141 |

26 Discretized Transforms and Sampling | 45 |

27 The Toy Example | 51 |

The Plancherel Transform for Locally Compact Groups | 59 |

32 Regularity Properties of Borel Spaces | 66 |

33 Direct Integrals | 67 |

332 Direct Integrals of von Neumann Algebras | 69 |

34 Direct Integral Decomposition | 71 |

342 Central Decompositions | 74 |

343 Type I Representations and Their Decompositions | 75 |

344 Measure Decompositions and Direct Integrals | 79 |

35 The Plancherel Transform for Unimodular Groups | 80 |

36 The Mackey Machine | 85 |

37 OperatorValued Integral Kernels | 93 |

38 The Plancherel Formula for Nonunimodular Groups | 97 |

52 Concrete Admissibility Conditions | 145 |

53 Concrete and Abstract Admissibility Conditions | 155 |

54 Wavelets on Homogeneous Groups | 160 |

55 Zak Transform Conditions for WeylHeisenberg Frames | 162 |

Sampling Theorems for the Heisenberg Group | 169 |

61 The Heisenberg Group and Its Lattices | 171 |

62 Main Results | 172 |

63 Reduction to WeylHeisenberg Systems | 174 |

64 WeylHeisenberg Frames | 176 |

65 Proofs of the Main Results | 178 |

66 A Concrete Example | 182 |

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191 | |

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### Common terms and phrases

admissibility conditions admissible vectors arbitrary Borel space Borel structure bounded compute construction continuous wavelet transform Corollary countable cyclic defined definition denote dense direct integral discrete series representations entails exists finite Fourier transform function G is unimodular given group G Haar measure Heisenberg group Hence Hilbert space Hilbert-Schmidt idempotent implies intertwining operator inversion formula irreducible representations isometry Jg Jg lattice Lebesgue measure Lemma Let G Lie group locally compact group Mackey mapping measurable field Moreover multiplicity Neumann algebras nonunimodular groups nonzero normalized tight frame Note obtain orthogonality particular Plancherel formula Plancherel inversion Plancherel measure Plancherel theorem Plancherel transform pointwise Proof Proposition quotient regular representation relation Remark sampling space scalar selfadjoint shows square-integrable subgroup subrepresentations subset theory tion topology unimodular groups unique unitary equivalence VNi(G von Neumann algebras Weyl-Heisenberg frames yields Zak transform