Orthogonal Systems and Convolution Operators

by Robert L. Ellis, Israel Gohberg

Overview

In this book we study orthogonal polynomials and their generalizations in spaces with weighted inner products. The impetus for our research was a deep theorem due to M.G. Krein along with subsequent results of Krein and H. Langer. Together with our colleagues, we have worked in this area for nearly fifteen years, and the results of our research are presented here in unified form. We are grateful to the Department of mathematics at the University of Maryland in College Park and to Tel-Aviv University for their support and encouragement. The support of the Silver Family Foundation is also highly appreciated. Introduction The starting point ofthis book is a study of the orthogonal polynomials {qn In—: O} obtained by orthogonalizing the power functions I, Z, z2, ... on the unit circle. The orthogonality is with respect to the scalar product defined by where the weight w is a positive integrable function on the unit circle. These ortho­ gonal polynomials are called the Szego polynomials associated with the weight w.

Product Details

ISBN-13:	9783034894180
Publisher:	Birkh�user Basel
Publication date:	10/24/2012
Series:	Operator Theory: Advances and Applications , #140
Edition description:	Softcover reprint of the original 1st ed. 2003
Pages:	238
Product dimensions:	6.10(w) x 9.25(h) x 0.02(d)

Table of Contents

1 Orthogonal Polynomials and Krein’s Theorem.- 2 Reformulations of Krein’s Theorem.- 3 Inner Products on Modules and Orthogonalization with Invertible Squares.- 4 Orthogonal Matrix Polynomials.- 5 Special Class of Block Toeplitz Matrices.- 6 Orthogonal Operator-Valued Polynomials: First Generalization.- 7 Convolution Equations on a Finite Interval.- 8 Continuous Analogues of Orthogonal Matrix Polynomials.- 9 Orthogonal Operator-Valued Polynomials.- 10 Reverse, Left and Right Orthogonalization.- 11 Discrete Infinite Analogue of Krein’s Theorem.- 12 Continuous Infinite Analogue of Krein’s Theorem.- References.- Index of Symbols.