Weyl Transforms 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. 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.