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 Fourier-Plancherel theory in $L^2(E_n)$. In addition to the general theory of integral transforms, connections are established with other areas of mathematical analysis - such as the theory of harmonic and analytic functions, the theory of orthogonal polynomials, and the moment problem - as well as to mathematical physics.
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 half-space 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 Paley-Wiener theorem Boundary properties of functions analytic in the upper half-plane 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 Wiener-Hopf technique.