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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.
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."
This scarce antiquarian book is included in our special Legacy
Reprint Series. In the interest of creating a more extensive
selection of rare historical book reprints, we have chosen to
reproduce this title even though it may possibly have occasional
imperfections such as missing and blurred pages, missing text, poor
pictures, markings, dark backgrounds and other reproduction issues
beyond our control. Because this work is culturally important, we
have made it available as a part of our commitment to protecting,
preserving and promoting the world's literature.
This scarce antiquarian book is included in our special Legacy
Reprint Series. In the interest of creating a more extensive
selection of rare historical book reprints, we have chosen to
reproduce this title even though it may possibly have occasional
imperfections such as missing and blurred pages, missing text, poor
pictures, markings, dark backgrounds and other reproduction issues
beyond our control. Because this work is culturally important, we
have made it available as a part of our commitment to protecting,
preserving and promoting the world's literature.
This scarce antiquarian book is included in our special Legacy
Reprint Series. In the interest of creating a more extensive
selection of rare historical book reprints, we have chosen to
reproduce this title even though it may possibly have occasional
imperfections such as missing and blurred pages, missing text, poor
pictures, markings, dark backgrounds and other reproduction issues
beyond our control. Because this work is culturally important, we
have made it available as a part of our commitment to protecting,
preserving and promoting the world's literature.
This scarce antiquarian book is included in our special Legacy
Reprint Series. In the interest of creating a more extensive
selection of rare historical book reprints, we have chosen to
reproduce this title even though it may possibly have occasional
imperfections such as missing and blurred pages, missing text, poor
pictures, markings, dark backgrounds and other reproduction issues
beyond our control. Because this work is culturally important, we
have made it available as a part of our commitment to protecting,
preserving and promoting the world's literature.
This scarce antiquarian book is included in our special Legacy
Reprint Series. In the interest of creating a more extensive
selection of rare historical book reprints, we have chosen to
reproduce this title even though it may possibly have occasional
imperfections such as missing and blurred pages, missing text, poor
pictures, markings, dark backgrounds and other reproduction issues
beyond our control. Because this work is culturally important, we
have made it available as a part of our commitment to protecting,
preserving and promoting the world's literature.
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