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Weyl transforms. (English) Zbl 0908.44002

Universitext. New York, NY: Springer. viii, 158 p. (1998).
From the author’s abstract: “The functional analytic properties of Weyl transforms as bounded linear operators on \(L^2(\mathbb{R}^n)\) 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 \(L^r(\mathbb{R}^{2n})\) and in terms of the Wigner transforms of Hermite functions are given. The roles of the Heisenberg group and 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 book consists of a preface and 30 chapters named as follows: Prerequisite topics in Fourier analysis; The Fourier-Wigner transform; The Wigner transform; The Weyl transform; Hilbert-Schmidt operators on \(L^2(\mathbb{R}^n)\); The tensor product in \(L^2(\mathbb{R}^n)\); \(H^*\)-algebras and the Weyl calculus; The Heisenberg group; The twisted convolution; The Riesz-Torin theorem; Weyl transform with symbols in \(L^r(\mathbb{R}^{2n})\), \(1\leq r\leq 2\); Weyl transforms with symbols in \(L^\infty(\mathbb{R}^{2n})\); Weyl transforms with symbols in \(L^r(\mathbb{R}^{2n})\), \(2\leq r\leq\infty\); Compact Weyl transforms; Localization operators; A Fourier transform; Compact localization operators; Hermite polynomials; Hermite functions; Laguerre polynomials; Hermite function on \(C\); Vector fields on \(C\); Laguerre formulas for Hermite functions on \(C\); Weyl transform on \(L^2(\mathbb{R}^n)\) with radial symbols; Another Fourier transform; A class of compact Weyl transforms on \(L^2(\mathbb{R})\); A class of bounded Weyl transforms on \(L^2(\mathbb{R})\); A Weyl transform with symbols in \(S(\mathbb{R}^2)\); The symplectic group; Symplectic invariance of Weyl transforms. The references contain 38 units.

MSC:

44A15 Special integral transforms (Legendre, Hilbert, etc.)
44-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to integral transforms
42-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to harmonic analysis on Euclidean spaces
43-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to abstract harmonic analysis
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