Laurent Series and their Padé Approximations

277 Pages·1987·9.6 MB·English

by  Adhemar Bultheel (auth.)

OT27:
Operator Theory: Advances and Applications
Vol. 27
Editor:
I. Gohberg
Tel Aviv University
Ramat-A viv, Israel
Editorial Office
School of Mathematical Sciences
Tel Aviv University
Ramat-A viv, Israel
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Editorial Board
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Birkhauser Verlag
Basel . Boston
Adhemar Bultheel
Laurent Series and their
Pade Approximations
1987 Birkhauser Verlag
Basel· Boston
Author's Address:
Adhemar Bultheel
K. U. Leuven
Dept. Computer Science
Celestijnenlaan 200A
3030 Leuven-Heverlee
Belgium
Library of Congress Cataloging in Publication Data
Bultheel, Adhemar:
Laurent series and the Pade approximations.
(Operator theory, advances and applications ; vol. 27)
Bibliography: p.
Includes index.
1. Laurent series. 2. Pade approximant. I. Title.
II. Series: Operator theory, advances and applications ; v. 27.
QA331.B795 1987 515 87-24297
ISBN-13: 978-3-0348-9988-8 e-ISBN-13: 978-3-0348-9306-0
DOl: 10.1007/978-3-0348-9306-0
CIP-Kurztitelaufnahme der Deutschen Bibliothek
Bultheel, Adhemar:
Laurent series and their Pade approximations /
Adhemar Bultheel. - Basel; Boston: Birkhauser,
1987.
(Operator theory; Vol. 27)
NE:GT
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© 1987 BirkhauserVeriag Basel
CONTENTS
Preface ix
2. Introduction 1
2.1 Classical Pade approximation 1
2.2 Toeplitz and Hankel systems 2
2.3 Continued fractions 3
2.4 Orthogonal polynomials 4
2.5 Rhombus algorithms and convergence 5
2.6 Block structure 5
2.7 Laurent-Pade approximants 6
2.8 The projection method 7
2.9 Applicati.o ns 7
2.10 Outline 10
3. Moebius transforms, continued fractions and Pade
approximants • 11
3.1 Moebius transforms 11
3.2 Flow graphs 14
3.3 Continued fractions (CF) 18
3.4 Formal series • 22
3.5 Pade approximants • 24
.
4. Two algorithms 29
4.1 Algorithm 1 29
4.2 Algorithm 2 32
5. All kinds of Pade Approximants 37
5.1 Pade approximants • 37
5.2 Laurent-Pade approximants 39
5.3 Two-point Pade approximants • 43
6. Continued fractions 47
6.1 General observations 47
6.2 Some special cases 49
7. Moebius transforms 55
7.1 General observations 55
7.2 Some special cases 57
8. Rhombus algorithms 65
vi CONTENTS
. . . . . . .
8.1 The ab parameters (sawtooth path) • . . . . . . . . 65
8.2 The FG parameters (row path) 72
8.3 A staircase path. • • • • • • • • • 73
8.4 po paramaters (diagonal path) •••••••••••• 75
8.5 Some dual results • • • • • • • • • • • • • • • 77
8.6 Relation with classical algorithms • • • • • • • • • • • 81
9. Biorthogonal polynomials, quadrature and reproducing
kernels • • • • • • • • • • • • • 83
9.1 Biorthogonal polynomials • • • • • • • • • • • • 83
9.2 Interpolatory quadrature methods • • • • • • • • • • • 90
9.3 Reproducing kernels • • • • • • • • • • 94
9.4 Other orthogonality relations • • • • • • • • • • • 98
10. Determinant expressions and matrix interpretations • • • • • 103
10.1 Determinant expressions. • 103
10.2 Matrix interpretations ••••••••••••• 112
10.2.1 Toeplitz matrices. 112
10.2.2 Hankel matrices • • 122
10.2.3 Tridiagonal matrices 127
11. Symmetry Properties • • • • • • 132
=
11.1 Symmetry for F(z) and F(z) F(1/ z) • • 132
11.2 Symmetry for F (z) and G (z) = 1/F (z) • • 136
12. Block structures. • • • • • • • • • • • • 141
12.1 Pade forms, Laurent-Pade forms and two-point Pade
forms • • • • • • • • • • • • • • • • . . . 141
12.2 The T-tab1e • • • • • • • • • • • • • • 143
12.3 The Pade, Laurent-Pade, and two-point Pade tables 149
13. Meromorphic functions and asymptotic behaviour. • • • • 155
13.1 The function F(z) • • • • • • • • • • • • • •• 155
13.2 Asymptotics for finite Toeplitz determinants ••••• 156
13.3 Asymptotics for infinite Toeplitz determinants • • • • • •• 159
13.4 Consequences for the T-tab1e • • • • • • • •• 163
14. Montessus de Ballore theorem for Laure.n t-.Pa.d e .. . . . . .
approximants. • • • • • • 167
14.1 Semi infinite Laurent series • 167
14.2 Bi-infinite Laurent series • 170
. . . . .
15. Determination of poles . . . 173
15.1 Rutishauser polynomials of type 1 and type 2 . . . 173
15.2 Rutishauser polynomials of type 3 . . 179
15.3 Rutishauser polynomials and Laurent series • • 181
CONTENTS vii
15.4 Convergence of parameters • 183
16. Determination of zeros 187
16.1 Dual Rutishauser polynomials and semi-infinite
series 187
16.2 From semi-infinite to bi-infinite series 189
16.3 Convergence of parameters • 193
17. Convergence in a row of the Laurent-Pade table 195
17.1 Toeplitz operators and the projection method 197
17.2 Convergence of the denominator 199
17.3 Convergence of the numerator • 203
18. The positive definite case and applications • 207
18.1 Function classes • 207
18.2 Connection with the previous results 212
18.3 Stochastic processes and systems • 219
18.4 Loss1ess inverse scattering and transmission lines 224
18.5 Laurent-Pade approximation and ARMA-filtering • 230
18.6 Concluding remarks 231
19. Examples 233
19.1 Example 1 • 233
19.2 Example 2 • 248
19.3 Example 3 • 253
References 257
List of symbols 263
Subject index 271
PREFACE
The Pade approximation problem is, roughly speaking, the local
approximation of analytic or meromorphic functions by rational ones. It is known
to be important to solve a large scale of problems in numerical analysis, linear
system theory, stochastics and other fields.
There exists a vast literature on the classical Pade problem. However,
these papers mostly treat the problem for functions analytic at 0 or, in a purely
algebraic sense, they treat the approximation of formal power series. For certain
problems however, the Pade approximation problem for formal Laurent series,
rather than for formal power series seems to be a more natural basis.
In this monograph, the problem of Laurent-Pade approximation is
central. In this problem a ratio of two Laurent polynomials in sought which
approximates the two directions of the Laurent series simultaneously.
As a side result the two-point Pade approximation problem can be solved. In that
case, two series are approximated, one is a power series in z and the other is a
power series in z-l. So we can approximate two, not necessarily different
functions one at zero and the other at infinity.
To connect this problem to the previous one, we just have to glue the two series
together to get a formal Laurent series. When the classical definition of Pade
approximation is applied to a Laurent series instead of a power series, we get a
numerator which is a polynomial plus infinitely many negative powers of z which
are generated by the terms with negative powers in the Laurent series. Thus if
the Laurent series has no negative powes of z, i.e. when it becomes a power series,
the classical Pade approximant appears as a special case.
The first part of this volume (up to chapter 11) is purely algebraic.
Three types of recursive algorithms are presented to find the solutions of the three
types of Pade problems mentioned above. The first one is in principle the
Trench-Zohar algorithm for Toeplitz matrix inversion. The second one is a
nonsymmetric version of the Schur algorithm for deciding whether a function is a
Schur function. Both these algorithms are based on a "Toeplitz approach", which
means that Toeplitz matrices are a basic working tool, whereas the algorithms
commonly used in the classical Pade literature (Euclid, Routh, Kronecker a.o) are
x
more "Hankel minded". A third type of algorithm is of rhombus type and this is
essentially the Rutishauser qd algorithm from numerical analysis.
From the engineering literature, the flow graph representation of
electrical networks is used to represent the algorithms. This will be most useful
in getting a visualization of the computational flow.
The recurrences of the algorithms can often be interpreted as
recurrences associated with continued fractions and formal orthogonal
polynomials. These interpretations are also treated to some extend, including
reproducing kernels and Christoffel-Darboux-type formulas.
When the algorithms are used in a linear algebra context, formulas will
be found for Toeplitz and Hankel matrix inversion and triangular factorization.
Finally, the block structure of the classical Pade table will be extended
to describe the blocks of Laurent-Pade and two-point Pade tables.
The second part of the volume is analytic in nature. The Pade
approximation problem is considered for meromorphic in C \ {oJ functions.
Classical and more recent results on the asymptotic behaviour of Hankel and
Toeplitz determinants and the projection method are used for proving convergence
in columns and rows of Pade tables. In classical Pade theory the convergence of
columns is well established. and the convergence of the rows can be obtained. from
a simple duality principle. For the Laurent-Pade problem, convergence of the
columns is essentially the same as in the classical case. For the row convergence
however, some new methods had to be used, and it is one of the most important
results of this monograph. From these convergence results it is derived how the
poles and zeros of a meromorphic function given by a Laurent series can be
computed. The computation of poles is related to column convergence and the
computation of zeros is related to row convergence.
Chapter l7 contains some interpretation of the Pade approximation
problem in other theories like Caratheooory and Schur function classes, Szego
polynomials orthogonal on the unit circle, prediction theory and inverse scattering.
My interest in the last topics was the inital motivation for writing this
monograph. Therefore it is my sincere hope that this text will not only be
appreciated by specialists in Pade approximation, continued fractions, orthogonal
polynomials and Toeplitz matrices but also by people with a more applied
mathematical background e.g. from stochastic processes, time series analysis,
signal processing, linear systems theory or inverse scattering.
I realize that it will take some time for the reader who is not familiar
with the subject to get accustomed to the large number of different types of
polynomials, series and parameters. To help him, a quick reference of the most
important formulas is included on a separate card which can be taken along as a
bookmarker.