Table Of ContentOperator Theory: Advances and
Applications
Vol. 140
Editor:
I. Gohberg
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Orthogonal Systems
and
Convolution Operators
Robert L. Ellis
Israel Gohberg
Springer Basel AG
Authors:
Robert L. Ellis Israel Gohberg
Department of Mathematics School of Mathematical Sciences
University of Maryland Raymond and Beverly Sackler
ColIege Park, MD 20742 Faculty of Exact Sciences
USA Tel Aviv University
e-mail: [email protected] IL - Ramat Aviv 69978
Israel
e-mail: [email protected]
2000 Mathematics Subject Classification 33C47, 42C05, 44A35, 47B35
A CIP catalogue record for this book is available from the
Library of Congress, Washington D.C., USA
Bibliographic information published by Die Deutsche Bibliothek
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in the Internet at <http://dnb.ddb.de>.
ISBN 978-3-0348-9418-0 ISBN 978-3-0348-8045-9 (eBook)
DOI 10.1007/978-3-0348-8045-9
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© 2003 Springer Basel AG
OriginalIy published by Birkhăuser Verlag iu 2003
Softcover reprint of the hardcover 1 st edition 2003
Printed on acid-free paper produced from chlorine-free pulp. TCF =
Cover design: Heinz Hiltbrunner, Basel
ISBN 978-3-0348-9418-0
987654321 www.birkhauser-science.com
In memory of M.G. Krein
Table of Contents
Preface.... ...................... ............ ...... .................. xi
Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. xiii
Chapter 1 Orthogonal Polynomials and Krein's Theorem............. 1
1.1 Szego's Theorem ................................................ 1
1.2 Krein's Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.3 Restatement of Krein's Theorem .................................. 13
1.4 One-Step Version of Krein's Theorem ............................. 15
1.5 Proof of Krein's Theorem ......................................... 23
1.6 Continuous Analogue of Krein's Theorem . . . . . . . . . . . . . . . . . . . . . . . . .. 25
1.7 Comments to Chapter 1 .......................................... 27
Chapter 2 Reformulations of Krein's Theorem. . . . . . . . . . . . . . . . . . . . . .. 29
2.1 Krein's Theorem in Terms ofInfinite Toeplitz Matrices .............. 29
2.2 Krein's Theorem in Terms of Modules ............................. 33
2.3 Comments to Chapter 2 .......................................... 35
Chapter 3 Inner Products on Modules and Orthogonalization
with Invertible Squares. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 37
3.1 Inner Products on Modules Over C* -Algebras ..................... 37
3.2 Orthogona1ization with Invertible Squares ......................... 40
3.3 Orthogonalization in the Module L;xr (1I') ......................... 46
3.4 Orthogonalization in the Module LR .............................. 48
3.5 Comments to Chapter 3 .......................................... 51
Chapter 4 Orthogonal Matrix Polynomials. . . . . . . . . . . . . . . . . . . . . . . . . .. 53
4.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 53
4.2 An Inertia Theorem ............................................. 55
4.3 Krein's Theorem for Matrix Polynomials. . . . . . . . . . . . . . . . . . . . . . . . .. 58
4.4 The Positive Definite Case ....................................... 63
4.5 Reformulation of Krein's Theorem for Matrix Polynomials .......... 65
4.6 Comments to Chapter 4 .......................................... 70
Chapter 5 Special Class of Block Toeplitz Matrices. . . . . . . . . . . . . . . . . .. 71
5.1 Preliminaries. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 72
5.2 An Identity ..................................................... 73
viii Table of Contents
5.3 Matrix Version of Rouche's Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 78
5.4 Special Form of Krein's Theorem ................................. 80
5.5 Example. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 85
5.6 Comments to Chapter 5 .......................................... 86
Chapter 6 Orthogonal Operator-Valued Polynomials:
First Generalization. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 87
6.1 Generalization of Krein's Theorem ................................ 87
6.2 Reformulation ofthe Main Theorem .............................. 93
6.3 Comments to Chapter 6 .......................................... 98
Chapter 7 Convolution Equations on a Finite Interval. . . . . . . . . . . . . . .. 99
7.1 Convolution Operators on a Finite Interval ......................... 99
7.2 Invertibility Conditions for Convolution Operators
on a Finite Interval ............................................. 101
7.3 The Inversion Formula .......................................... 106
7.4 Comments to Chapter 7 ......................................... 114
Chapter 8 Continuous Analogues of Orthogonal
Matrix Polynomials. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 115
8.1 The Main Theorem............................................. 115
8.2 An Identity .................................................... 117
8.3 Discretization. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 122
8.4 Wiener-Hopf Operators ......................................... 129
8.5 Proof of the Main Theorem ...................................... 132
8.6 Reformulation of Krein's Theorem for Continuous
Analogues of Orthogonal Matrix Polynomials ..................... 139
8.7 Examples of Continuous Analogues of Orthogonal
Matrix Polynomials ............................................. 140
8.8 Comments to Chapter 8 ......................................... 142
Chapter 9 Orthogonal Operator-Valued Polynomials. . . . . . . . . . . . . . .. 143
9.1 Generalized Inertia Theorems.. . . . . . . .. . . .. . . . . . . ... . ... . .. . . . .. 143
9.2 Orthogonal Operator-Valued Polynomials........................ 148
9.3 Comments to Chapter 9 ......................................... 158
Appendix to Chapter 9 Time-Variant Analogues of Orthogonal
Polynomials. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 159
A.l Inertia Theorem. . . . . . . . . . . . . ... . . .. . . . . . . . .. . . . . . . ... . .. . . . . . .. 159
A.2 Time-Variant Analogue of Krein's Theorem ...................... 160
A.3 Comments to the Appendix..................................... 161
Table of Contents ix
Chapter 10 Reverse, Left and Right Orthogonalization .............. 163
10.1 Reverse Orthogonalization with Respect to a Scalar Product . . . . . .. 163
10.2 Left and Right Orthogonalization of Two-Sided Sequences ........ 173
10.3 Reverse Orthogonalization with Respect to an Inner Product ...... 176
10.4 Comments to Chapter 10 ...................................... 182
Chapter 11 Discrete Infinite Analogue of Krein's Theorem. . . . . . . . .. 183
11.1 Preliminaries. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 184
11.2 Left and Right Orthogonalization ............................... 189
11.3 Identities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 194
11.4 Analogue of Krein's Theorem .................................. 198
11.5 Comments to Chapter 11 ...................................... 203
Chapter 12 Continuous Infinite Analogue of Krein's Theorem ....... 205
12.1 Statement of Main Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 205
12.2 Auxilliary Theorems .......................................... 208
12.3 Proof of Main Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 218
12.4 Comments to Chapter 12 ...................................... 226
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 227
Index of Symbols. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233
Index ............................................................... 235
Preface
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.
Up to constant factors, the polynomials are given by
to LI Ln
tl to Ln+1
qn(Z) = det (n ?: 1)
tn-I tn-2 LI
1 z zn
where the numbers Ln, Ln+ I, ... , tn-I are moments of the weight w:
j1f
tt=_l e-W)w(eiB)d() (.e=O,±I,±2, ... )
2rr
-1f
The Szego polynomials can also be obtained, up to constant factors, by letting
where ao, ... , an satisfy the equation
(1.1)
=
and Tn (tj-dJ,k=O is the (n+ 1) x (n+ 1) selfadjoint Toeplitz matrix determined
by the moments Ln, ... , tn. This characterization of the Szego polynomials is very
convenient for later generalizations. G. Szego proved that all the zeros ofthe Szego
polynomials lie inside the unit circle.
M.G. Krein studied orthogonal polynomials arising from the orthogonalization
of 1, Z, Z2, •.. with respect to an indefinite scalar product with a weight w that need
