Discrepancy of Signed Measures and Polynomial Approximation

444 Pages·2002·10.05 MB·English

by  Vladimir V. Andrievskii

Springer Monographs in Mathematics
Springer Science+Business Media, LLC
Vladimir V. Andrievskii Hans-Peter Blatt
Discrepancy of Signed
Measures and Polynomial
Approximation
, Springer
Vladimir V. Andrievskii
Department of Mathematics and Computer Science
Kent State University
Kent, OH 44242
USA
[email protected]
Hans-Peter Blatt
Mathematisch-Geographische Fakultät
Katholische Universität Eichstätt
D-850n Eichstätt
Germany
hans. [email protected]
Mathematics Subject Classification (2000): 26CI0, 31 A15, 30E I 0, 42C05, 4
Library ofCongress Cataloging-in-Publication Data
Andrievskii, V. V., 1953-
Discrepancy of signed measures and polynomial approximation 1 Vladimir V.
Andrievskii, Hans-Peter Blatt.
p. cm.-(Springer monographs in mathematics)
Inc1udes bibliographical references and index.
ISBN 978-1-4419-3146-7 ISBN 978-1-4757-4999-I(eBook)
DOI 10.1007/978-1-4757-4999-1
1. Approximation theory. 2. Orthogonal polynomials. l. Blatt, Hans-Peter. II. Title. III.
Series.
QA221 .A3567 2001
511.~c21 2001032837
Printed on acid-free paper.
© 2002 Springer Science+Business Media New Y ork
Originally published by Springer-Verlag New York, Inc. in 2002
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ISBN 978-1-4419-3146-7 SPIN 10696722
To Elena
Olga
Sophie
Sarah
Simon
Markus
Preface
In many situations in approximation theory the distribution of points in
a given set is of interest. For example, the suitable choiee of interpolation
points is essential to obtain satisfactory estimates for the convergence of
interpolating polynomials. Zeros of orthogonal polynomials are the nodes
for Gauss quadrat ure formulas. Alternation points of the error curve char
acterize the best approximating polynomials.
In classieal complex analysis an interesting feature is the location of zeros
of approximants to an analytie function. In 1918 R. Jentzsch [91] showed
that every point of the circle of convergence of apower series is a limit
point of zeros of its partial sums. This theorem of Jentzsch was sharpened
by Szegö [170] in 1923. He proved that for apower series with finite radius
of convergence there is an infinite sequence of partial sums, the zeros of
whieh are "equidistributed" with respect to the angular measure.
In 1929 Bernstein [27] stated the following theorem. Let f be a positive
continuous function on [-1, 1]; if almost all zeros of the polynomials of best
approximation to f (in a weighted L2-norm) are outside of an open ellipse
c with foci at -1 and 1, then f has a continuous extension that is analytic
c.
in
Results extremely useful for the study of "equidistribution" were pub
lished by Erdös and Thnin [55, 57], who established estimates for the distri
bution of zeros of a monie polynomial the uniform norm of whieh is known
on the interval [-1, 1] or on the unit disko
From another point of view, the theorem of Jentzsch tells us that every
boundary point of the domain of uniform convergence of the partial sums of
apower series is a limit point of zeros of partial sums. This, of course, is not
Vlll Preface
true for general sequences of polynomials. Ostrowski [136] and Szegö [169]
have given supplementary sufficient conditions to ensure that the boundary
points of the domain of uniform convergence for a sequence of polynomials
are limit points of their zeros.
Walsh [181] considered, among other things, interpolation points on the
boundary of a set E where the function f has to be approximated by
polynomials. He discovered that the distribution of the interpolation points
has to be related to the equilibrium distribution of E to obtain sufficiently
good approximations.
In 1963 Kadec [92] proved the surprising result that the distribution
of alternation points in Chebyshev polynomial approximation on areal
interval is related to its equilibrium measure.
Each of the results just mentioned has been generalized in different di
rections and has become a source of many fruitful constructions in func
tion theory. The crucial key to these generalizations can be found in a
potential-theoretical interpretation of the classical results above. There is
a huge literat ure dealing with this field. We cite here only the main re
sults directly related to the subject und er consideration. The topics of this
book reflect the authors' personal mathematical interests and point of view
on key ideas and results concerning Erdös-Turan and Jentzsch-Szegö type
theorems and their applications.
The subjects and goals of the book are apparent from the table of con
tents. To our regret, it was not possible to touch quest ions related to the
distribution of zeros, poles, and extreme points in complex rational approx
imation; generalizations of Ostrowski gaps in power series; Tsuji points;
Leja points; zeros of polynomials of best approximation to finitely and in
finitely differentiable functions; or to consider zeros and values of Faber
polynomials.
At the end of the main chapters we have included a section entitled
"Historical Comments," which contains citations for many of the theorems.
It would be presumptuous to trace a complete history of the ideas presented
in this book. Nevertheless, in general, the absence of a reference implies
originality on our part, to our best knowledge.
Our main tool is the application of some basic not ions and facts from
potential theory and conformal invariants, such as the module of a family
of curves. Moreover, some methods and results from the theory of qua
siconformal mappings in the complex plane will play an important roIe,
too.
Mario Götz and Jörg Hüsing have carefully studied the manuscript and
have contributed various corrections and improvements to the presentation.
We would Iike to extend to them our deepest gratitude. Further, we want
to thank Wolfgang Schmidt for drawing the figures.
To D. Gaier, R. Grothmann, H.N. Mhaskar, Ch. Pommerenke, E.B. Saff,
H. Stahl, and V. Totik we extend our sincere appreciation for their encour
agement and heipful comments.
Preface ix
Thanks go to Katharina Surovcik for typing part of the manuscript, as
weH as to Markus and Simon for their great support to tie the figures into
the text and to put together the final manusc:dpt.
Last but not least we want to thank Ina Lindemann and the editorial staff
at Springer-Verlag for their continuous support and unending patience.
Eichstätt, Kent V. V. Andrievskii, H.-P. Blatt
Contents
Preface vii
1 A uxiliary Facts 1
1.1 Basie Potential-Theoretieal Coneepts . 1
1.1.1 Sets and Curves ....... . 1
1.1.2 Harmonie and Subharmonie Funetions,
Maximum Principle 2
1.1.3 Green's Formula ............ . 6
1.1.4 Harmonie Measure . . . . . . . . . . .. 8
1.1.5 Energy, Capacity, Equilibrium Measure 10
1.1.6 Maximum Principle ....... . 14
1.1.7 Weak* Topology, Helly's Theorem 14
1.1.8 Green's Function .......... . 14
1.1.9 Fekete Points, Transfinite Diameter,
Chebyshev Constant . . . . . . . . . 17
1.2 Conformal and Quasieonformal Mappings 19
1.2.1 Basic Principles of Complex Analysis. 19
1.2.2 Equilibrium Measure and Conformal Mappings 21
1.2.3 Distortion Theorems. . . . . . . . . . . 22
1.2.4 Module of a Family of Curves and Ares 24
1.2.5 Quasieonformal Curves and Ares . . . 27
1.2.6 Dini-Smooth Curves and Ares. . . . . 32
1.3 Faber Polynomials and Grunsky Coefficients . 37
1.4 Monie ~olynomials . . . . . . . . . 40
1.5 Linear Approximation . . . . . . . 44
1.5.1 Polynomial Approximation 45
1.5.2 Averaging and Smoothing 46
1.6 Ordering Symbols .. 48
1. 7 Historieal Comments . . . . . . . 48
xii Contents
2 Zero Distribution of Polynomials 49
2.1 Jentzseh-Szegö Type Theorems. . . . . . . . . . . . . .. 50
2.2 Erdös-Tunin Type Theorems for Quasieonformal Curves
and Ares. . . . . . . . . . . . . . . . . . . . . . . . . . 56
2.2.1 Polynomial Bounds on Quasiconformal Curves .. 57
2.2.2 Polynomial Bounds on Quasiconformal Ares. . .. 68
2.3 Sharpness and Extensions of Erdös-Turan Type Theorems. 72
2.4 Erdös-Turan Type Theorems on Compact Sets with
Smooth Curves and Ares. . . . . . . . . . . . . . . . . 78
2.5 Erdös-Turan Type Theorems on a System of Intervals 86
2.6 Historical Comments . . . . . . . . . . . . . . . . . . 92
3 Discrepancy Theorems via Two-Sided Bounds for
Potentials 95
3.1 Estimates for Quasiconformal Curves and Ares . . 96
3.2 Loeal Estimates for Intervals ............ 108
3.3 Loeal Estimates for Dini-Smooth Ares and Curves 114
3.4 Historical Comments . . . . . . . . . . . .' . . . . . 128
4 Discrepancy Theorems via One-Sided Bounds for
Potentials 129
4.1 Outer Bounds for Potentials . . . . . . . . . . . . . . 130
4.2 Inner and Outer Bounds for Potentials . . . . . . . . 150
4.3 Inner Bounds for Potentials of Signed Measures on Analytic
Jordan Curves ............... 152
4.4 Another Approach for Dini-Smooth Ares. 161
4.5 Historical Comments . . . . . . . . . . . . 172
5 Discrepancy Theorems via Energy Integrals 173
5.1 Modulus of a Doubly Split Plane . . . . . . . 174
5.2 Estimates of Mass Distributions from Their Energies 178
5.3 Historical Comments . . . . . . . . . . . . . . . . . . 186
6 Applications of Jentzsch-Szegö and Erdös-Turan Type
Th~~ms 1M
6.1 Polynomials of Best Uniform Approximation 188
6.2 Polynomials of Near-Best Approximation .. 193
6.3 Polynomials of Maximal Convergenee. . . . . 202
6.4 a-Values of Orthogonal Polynomials on Quasidisks 208
6.5 Zeros Qf Bieberbaeh Polynomials and Their Derivatives. 215
6.6 Historical Comments . . . . . . . . . . . . . . . . . . . . 223
7 Applications of Discrepancy Theorems 225
7.1 Distribution of Fekete Points ..... . 226
7.2 Fekete Points for Domains with Analytie Boundary . 229