The functional analytic properties of Weyl transforms as bounded
linear operators on $ LA2A1/4(ABbb RA1/4AnA1/4) $ 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 $ LArA1/4(ABbb RA1/4A2nA1/4) $ 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.
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