ELEMENTS OF ABSTRACT HARMONIC ANALYSIS

By D. NOERU

CONTENTS INTRODUCTION THE POINT OF DEPARTURE OVERVIEW OF THE BOOK PRELIMINARIES WAVELET TRANSFORMS AND GROUP REPRESENTATIONS HAAR MEASURE AND THE REGULAR REPRESENTATION COHERENT STATES AND RESOLUTIONS OF THE IDENTITY CONTINUOUS WAVELET TRANSFORMS AND THE REGULAR REPRESENTATION DISCRETE SERIES REPRESENTATIONS SELFADJOINT CONVOLUTION IDEMPOTENTS AND SUPPORT PROPERTIES DISCRETIZED TRANSFORMS AND SAMPLING THE TOY EXAMPLE THE PLANCHEREL TRANSFORM FOR LOCALLY COMPACT GROUPS A DIRECT INTEGRAL VIEW OF THE TOY EXAMPLE REGULARITY PROPERTIES OF BOREL SPACES? DIRECT INTEGRALS DIRECT INTEGRALS OF HILBERT SPACES DIRECT INTEGRALS OF VON NEUMANN ALGEBRAS DIRECT INTEGRAL DECOMPOSITION CENTRAL DECOMPOSITIONS? TYPE I REPRESENTATIONS AND THEIR DECOMPOSITIONS MEASURE DECOMPOSITIONS AND DIRECT INTEGRALS THE PLANCHEREL TRANSFORM FOR UNIMODULAR GROUPS THE MACKEY MACHINE? OPERATOR-VALUED INTEGRAL KERNELS? THE PLANCHEREL FORMULA FOR NONUNIMODULAR GROUPS THE PLANCHEREL THEOREM CONSTRUCTION DETAILS? PLANCHEREL INVERSION AND WAVELET TRANSFORMS FOURIER INVERSION AND THE FOURIER ALGEBRA? PLANCHEREL INVERSION? ADMISSIBILITY CRITERIA ADMISSIBILITY CRITERIA AND THE TYPE I CONDITION?? WIGNER FUNCTIONS ASSOCIATED TO NILPOTENT LIE GROUPS?? ADMISSIBLE VECTORS FOR GROUP EXTENSIONS QUASIREGULAR REPRESENTATIONS AND THE DUAL ORBIT SPACE CONCRETE ADMISSIBILITY CONDITIONS CONCRETE AND ABSTRACT ADMISSIBILITY CONDITIONS WAVELETS ON HOMOGENEOUS GROUPS?? ZAK TRANSFORM CONDITIONS FOR WEYL-HEISENBERG FRAMES SAMPLING THEOREMS FOR THE HEISENBERG GROUP THE HEISENBERG GROUP AND ITS LATTICES MAIN RESULTS REDUCTION TO WEYL-HEISENBERG SYSTEMS? WEYL-HEISENBERG FRAMES? PROOFS OF THE MAIN RESULTS? A CONCRETE EXAMPLE REFERENCES