Ebook: Orthogonal Systems and Convolution Operators

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 ofthe 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.

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The main concern of this book is the distribution of zeros of polynomials that are orthogonal on the unit circle with respect to an indefinite weighted scalar or inner product. The first theorem of this type, proved by M. G. Krein, was a far-reaching generalization of G. Szeg?'s result for the positive definite case. A continuous analogue of that theorem was proved by Krein and H. Langer. These results, as well as many generalizations and extensions, are thoroughly treated in this book. A unifying theme is the general problem of orthogonalization with invertible squares in modules over C*-algebras. Particular modules that are considered in detail include modules of matrices, matrix polynomials, matrix-valued functions, linear operators, and others. One of the central features of this book is the interplay between orthogonal polynomials and their generalizations on the one hand, and operator theory, especially the theory of Toeplitz marices and operators, and Fredholm and Wiener-Hopf operators, on the other hand. The book is of interest to both engineers and specialists in analysis.

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The main concern of this book is the distribution of zeros of polynomials that are orthogonal on the unit circle with respect to an indefinite weighted scalar or inner product. The first theorem of this type, proved by M. G. Krein, was a far-reaching generalization of G. Szeg?'s result for the positive definite case. A continuous analogue of that theorem was proved by Krein and H. Langer. These results, as well as many generalizations and extensions, are thoroughly treated in this book. A unifying theme is the general problem of orthogonalization with invertible squares in modules over C*-algebras. Particular modules that are considered in detail include modules of matrices, matrix polynomials, matrix-valued functions, linear operators, and others. One of the central features of this book is the interplay between orthogonal polynomials and their generalizations on the one hand, and operator theory, especially the theory of Toeplitz marices and operators, and Fredholm and Wiener-Hopf operators, on the other hand. The book is of interest to both engineers and specialists in analysis.
Content:
Front Matter....Pages i-xvi
Orthogonal Polynomials and Krein’s Theorem....Pages 1-27
Reformulations of Krein’s Theorem....Pages 29-36
Inner Products on Modules and Orthogonalization with Invertible Squares....Pages 37-51
Orthogonal Matrix Polynomials....Pages 53-70
Special Class of Block Toeplitz Matrices....Pages 71-86
Orthogonal Operator-Valued Polynomials: First Generalization....Pages 87-98
Convolution Equations on a Finite Interval....Pages 99-114
Continuous Analogues of Orthogonal Matrix Polynomials....Pages 115-142
Orthogonal Operator-Valued Polynomials....Pages 143-161
Reverse, Left and Right Orthogonalization....Pages 163-182
Discrete Infinite Analogue of Krein’s Theorem....Pages 183-203
Continuous Infinite Analogue of Krein’s Theorem....Pages 205-226
Back Matter....Pages 227-236

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The main concern of this book is the distribution of zeros of polynomials that are orthogonal on the unit circle with respect to an indefinite weighted scalar or inner product. The first theorem of this type, proved by M. G. Krein, was a far-reaching generalization of G. Szeg?'s result for the positive definite case. A continuous analogue of that theorem was proved by Krein and H. Langer. These results, as well as many generalizations and extensions, are thoroughly treated in this book. A unifying theme is the general problem of orthogonalization with invertible squares in modules over C*-algebras. Particular modules that are considered in detail include modules of matrices, matrix polynomials, matrix-valued functions, linear operators, and others. One of the central features of this book is the interplay between orthogonal polynomials and their generalizations on the one hand, and operator theory, especially the theory of Toeplitz marices and operators, and Fredholm and Wiener-Hopf operators, on the other hand. The book is of interest to both engineers and specialists in analysis.
Content:
Front Matter....Pages i-xvi
Orthogonal Polynomials and Krein’s Theorem....Pages 1-27
Reformulations of Krein’s Theorem....Pages 29-36
Inner Products on Modules and Orthogonalization with Invertible Squares....Pages 37-51
Orthogonal Matrix Polynomials....Pages 53-70
Special Class of Block Toeplitz Matrices....Pages 71-86
Orthogonal Operator-Valued Polynomials: First Generalization....Pages 87-98
Convolution Equations on a Finite Interval....Pages 99-114
Continuous Analogues of Orthogonal Matrix Polynomials....Pages 115-142
Orthogonal Operator-Valued Polynomials....Pages 143-161
Reverse, Left and Right Orthogonalization....Pages 163-182
Discrete Infinite Analogue of Krein’s Theorem....Pages 183-203
Continuous Infinite Analogue of Krein’s Theorem....Pages 205-226
Back Matter....Pages 227-236
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