Synopses & Reviews
A study of the functional analytic properties of Weyl transforms as bounded linear operators on $ L2ü(äBbb Rünü) $ in terms of the symbols of the transforms. Further, the boundedness, the compactness, the spectrum and the functional calculus of the Weyl transform are proved in detail, while new results and techniques on the boundedness and compactness of the Weyl transforms in terms of the symbols in $ Lrü(äBbb Rü2nü) $ and in terms of the Wigner transforms of Hermite functions are given. The roles of the Heisenberg group and the symplectic group in the study of the structure of the Weyl transform are explained, and the connections of the Weyl transform with quantization are highlighted throughout the book. Localisation operators, first studied as filters in signal analysis, are shown to be Weyl transforms with symbols expressed in terms of the admissible wavelets of the localisation operators. The results and methods mean this book is of interest to graduates and mathematicians working in Fourier analysis, operator theory, pseudo-differential operators and mathematical physics.
Synopsis
This book is an outgrowth of courses given by me for graduate students at York University in the past ten years. The actual writing of the book in this form was carried out at York University, Peking University, the Academia Sinica in Beijing, the University of California at Irvine, Osaka University, and the University of Delaware. The idea of writing this book was ?rst conceived in the summer of 1989, and the protracted period of gestation was due to my daily duties as a professor at York University. I would like to thank Professor K. C. Chang, of Peking University; Professor Shujie Li, of the Academia Sinica in Beijing; Professor Martin Schechter, of the University of California at Irvine; Professor Michihiro Nagase, of Osaka University; and Professor M. Z. Nashed, of the University of Delaware, for providing me with stimulating environments for the exchange of ideas and the actual writing of the book. We study in this book the properties of pseudo-differential operators arising in quantum mechanics, ?rst envisaged in 33] by Hermann Weyl, as bounded linear 2 n operators on L (R ). Thus, it is natural to call the operators treated in this book Weyl transforms.
Synopsis
This is a compact, well organized treatment of a part of harmonic analysis that has important applications in mathematical physics. It studies properties of pseudo-differential operators which arise in quantum mechanics. The book is self-contained and no familiarity with pseudo-differential operators is required for a good understanding of the material.
Table of Contents
* Prerequisite Topics in Fourier Analysis * The Fourier-Wigner Transform * The Wigner Transform * The Weyl Transform * Hilbert- Schmidt Operators on L2(IRn) * The Tensor Product in L2(IRn) * H* Algebras and the Weyl Calculus * The Heisenberg Group * The Twisted Convolution * The Riesz-Thorin Theorem * Weyl Transforms with Symbols in LT(IR2n), 1 l1 r l2 * Weyl Transforms with Symbols in L?(IR2n) * Weyl Transforms with Symbols in LT(IR2n), 1 l1 r l2 * Compact Weyl Transforms * Localization Operators * A Fourier Transform * Compact Localization Operators * Hermite Polynomials * Hermite Functions * Laguerre Polynomials * Hermite Functions on C * Vector Fields on C * Laguerre Formulas for Hermite Functions on C * Weyl Transforms on L2(IR) with Radial Symbols * Another Fourier Transform * A Class of Compact Weyl Transforms on L2(IR) * A Class of Bounded Weyl Transforms on L2(IR) * A Weyl Transform with Symbol in S' /IR 2 * The Symplectic Group * Symplectic Invariance of Weyl Transforms