## OP-SF WEB

### Extract from OP-SF NET


Topic #6  ------------   OP-SF NET 7.6  -------------   November 15, 2000
~~~~~~~~~~~~~
From: Tom Koornwinder (thk@science.uva.nl)
Subject: New book on Fourier Series in Orthogonal Polynomials

FOURIER SERIES IN ORTHOGONAL POLYNOMIALS
by  Boris Osilenker (Moscow State Civil Engineering University)
World Scientific, 1999

>From the web site:

http://www.wspc.com.sg/books/mathematics/4039.htm

This book presents a systematic course on general orthogonal polynomials and
Fourier series in orthogonal polynomials. It consists of six chapters. Chapter 1
deals in essence with standard results from the university course on the function
theory of a real variable and on functional analysis. Chapter 2 contains the
classical results about the orthogonal polynomials (some properties, classical
Jacobi polynomials and the criteria of boundedness).

The main subject of the book is Fourier series in general orthogonal polynomials.
Chapters 3 and 4 are devoted to some results in this topic (classical results
about convergence and summability of Fourier series in L2m; summability almost
everywhere by the Cesaro means and the Poisson-Abel method for Fourier polynomial
series are the subject of Chapters 4 and 5).

The last chapter contains some estimates regarding the generalized shift operator
and the generalized product formula, associated with general orthogonal
polynomials.

The starting point of the technique in Chapters 4 and 5 is the representations of
bilinear and trilinear forms obtained by the author. The results obtained in
these two chapters are new ones.

Chapters 2 and 3 (and part of Chapter 1) will be useful to postgraduate students,
and one can choose them for treatment.

This book is intended for researchers (mathematicians, mechanicians and
physicists) whose work involves function theory, functional analysis, harmonic
analysis and approximation theory.

Contents:
Orthogonal Polynomials and Their Properties
Convergence and Summability of Fourier Series in L^2_\mu
Fourier Orthogonal Series in L^r_\mu (1 < r < \infty) and C
Fourier Polynomial Series in L^1_\mu. Analogs of Fatou Theorems
The Representations of the Trilinear Kernels. Generalized
Translation Operator in Orthogonal Polynomials

Readership: Researchers in mathematics, mechanics and physics involved in
function theory, functional analysis, harmonic analysis and approximation
theory.

296pp    Pub. date: Apr 1999
ISBN 981-02-3787-1   US\$55 / È34