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Product Code:  MMONO/70 
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Softcover ISBN:  9780821845240 
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Softcover ISBN:  9780821845240 
Product Code:  MMONO/70 
List Price:  $165.00 
MAA Member Price:  $148.50 
AMS Member Price:  $132.00 
eBook ISBN:  9781470444846 
Product Code:  MMONO/70.E 
List Price:  $155.00 
MAA Member Price:  $139.50 
AMS Member Price:  $124.00 
Softcover ISBN:  9780821845240 
eBook ISBN:  9781470444846 
Product Code:  MMONO/70.B 
List Price:  $320.00 $242.50 
MAA Member Price:  $288.00 $218.25 
AMS Member Price:  $256.00 $194.00 

Book DetailsTranslations of Mathematical MonographsVolume: 70; 1988; 108 ppMSC: Primary 44
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 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.

Table of Contents

Chapters

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


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


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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 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.

Chapters

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

“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