Translations of Mathematical Monographs 1988; 108 pp; softcover Volume: 70 Reprint/Revision History: reprinted 1992 ISBN10: 0821845241 ISBN13: 9780821845240 List Price: US$45 Member Price: US$36 Order Code: MMONO/70
 This book, which grew out of lectures given over the course of several years at Kharkov University for students in the Faculty of Mechanics and Mathematics, is devoted to classical integral transforms, principally the Fourier transform, and their applications. The author develops the general theory of the Fourier transform for the space \(L^1(E_n)\) of integrable functions of \(n\) variables. His proof of the inversion theorem is based on the general Bochner theorem on integral transforms, a theorem having other applications within the subject area of the book. The author also covers FourierPlancherel theory in \(L^2(E_n)\). In addition to the general theory of integral transforms, connections are established with other areas of mathematical analysissuch as the theory of harmonic and analytic functions, the theory of orthogonal polynomials, and the moment problemas well as to mathematical physics. Reviews ""This book is remarkable for its rigor, brevity, and systematic expression which, together with the problems proposed in each chapter, make it extremely useful for students, mathematicians, and physicists.""  Mathematical Reviews Table of Contents  Averaging operators and the Bochner theorem
 The Fourier transform in \(L^1\)
 The inversion theorem in \(L^1\). The Poisson integral
 Harmonic functions. The Dirichlet problem for a ball and a halfspace
 The Fourier transform in \(L^2\)
 Hermite functions
 Spherical functions
 Positive definite functions
 The Hankel transform
 Orthogonal polynomials and the moment problem
 The class \(H^2\). The PaleyWiener theorem
 Boundary properties of functions analytic in the upper halfplane and the Hilbert transform
 The Poisson summation formula and some of its applications
 Applications of the Laplace and Fourier transforms to the solution of boundary value problems in mathematical physics
 Fourier transforms of increasing functions. The WienerHopf technique
