Weyl Transforms (Universitext) - Hardcover

Wong, M.W.

Hardcover

ISBN 10: 0387984143 ISBN 13: 9780387984148

Publisher: Springer, 1998

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.

About this title

Synopsis

The functional analytic properties of Weyl transforms as bounded linear operators on $ L^a2u(aBbb Ru^anu) $ are studied in terms of the symbols of the transforms. The boundedness, the compactness, the spectrum and the functional calculus of the Weyl transform are proved in detail. New results and techniques on the boundedness and compactness of the Weyl transforms in terms of the symbols in $ L^aru(aBbb Ru^a2nu) $ 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 explicated, and the connections of the Weyl transform with quantization are highlighted throughout the book. Localization 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 localization operators. The results and methods in this book should be of interest to graduate students and mathematicians working in Fourier analysis, operator theory, pseudo- differential operators and mathematical physics.

Background materials are given in adequate detail to enable a graduate student to proceed rapidly from the very basics to the frontier of research in an area of operator theory.

Edition details

• Publisher: Springer
• Publication date: 1998
• Language: English
• ISBN 10: 0387984143
• ISBN 13: 9780387984148
• Binding: Hardcover
• Number of pages: 168