Weyl Transforms

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.

Contenu

Prerequisite Topics in Fourier Analysis.- The Fourier-Wigner Transform.- The Wigner Transform.- The Weyl Transform.- Hilbert-Schmidt Operators on L2(?n).- The Tensor Product in L2(?n).- H*-Algebras and the Weyl Calculus.- The Heisenberg Group.- The Twisted Convolution.- The Riesz-Thorin Theorem.- Weyl Transforms with Symbols in Lr(?2n), 1 ? r ? 2.- Weyl Transforms with Symbols in L?(?2n).- Weyl Transforms with Symbols in Lr(?2n), 2 r < ?.- Compact Weyl Transforms.- Localization Operators.- A Fourier Transform.- Compact Localization Operators.- Hermite Polynomials.- Hermite Functions.- Laguerre Polynomials.- Hermite Functions on ?.- Vector Fields on ?.- Laguerre Formulas for Hermite Functions on ?.- Weyl Transforms on L2(?) with Radial Symbols.- Another Fourier Transform.- A Class of Compact Weyl Transforms on L2(?).- A Class of Bounded Weyl Transforms on L2(?).- A Weyl Transform with Symbol in S'(?2).- The Symplectic Group.- Symplectic Invariance of Weyl Transforms.