**Translations of Mathematical Monographs**

1988;
108 pp;
Softcover

MSC: Primary 44;

Print ISBN: 978-0-8218-4524-0

Product Code: MMONO/70

List Price: $48.00

AMS Member Price: $38.40

MAA Member Price: $43.20

**Electronic ISBN: 978-1-4704-4484-6
Product Code: MMONO/70.E**

List Price: $48.00

AMS Member Price: $38.40

MAA Member Price: $43.20

# Lectures on Integral Transforms

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*N. I. Akhiezer*

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.

#### Reviews & Endorsements

“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

# Table of Contents

## Lectures on Integral Transforms

- Cover Cover11
- Title page iii4
- Contents v6
- Averaging operators and the Bochner theorem 18
- The Fourier transform in 𝐿¹ 916
- The inversion theorem in 𝐿¹. The Poisson integral 1623
- Harmonic functions. The Dirichlet problem for a ball and a half-space 2431
- The Fourier transform in 𝐿² 3340
- Hermite functions 3946
- Spherical functions 4249
- Positive definite functions 5259
- The Hankel transform 5966
- Orthogonal polynomials and the moment problem 6471
- The class 𝐻². The Paley-Wiener theorem 7279
- Boundary properties of functions analytic in the upper half-plane and the Hilbert transform 8390
- The Poisson summation formula and some of its applications 8996
- Applications of the Laplace and Fourier transforms to the solution of boundary value problems in mathematical physics 94101
- Fourier transforms of increasing functions. The Wiener-Hopf technique 100107
- Back Cover Back Cover1117