Weighted Approximation with Varying Weight
A new construction is given for approximating a logarithmic potential by a discrete one. This yields a new approach to approximation with weighted polynomials of the form w"n"(" "= uppercase)P"n"(" "= uppercase). The new technique settles several open pro
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1569
Vilmos Totik
WeightedApproximation with VaryingWeight
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Author Vilmos Totik Bolyai Institute University of Szeged Aradi v. tere 1 6720 Szeged, Hungary and Department of Mathematics University of South Florida Tampa, FL 33620, USA
Mathematics Subject Classification (1991): 41A10, 41A17, 41A25, 26Cxx, 31A10, 31A99, 41A21, 41A44, 42C05, 45E05
ISBN 3-540-57705-X Springer-Verlag Berlin Heidelberg New York ISBN 0-387-57705-X Springer-Verlag New York Berlin Heidelberg Library of Congress Cataloging-in-Publication Data. Totik, V. Weighted approximation with varying weights/Vilmos Totik. p. cm. - (Lecture notes in mathematics; 1569) Includes bibliographical references and index. ISBN 3-540-57705-X (Berlin: softcover: acid-free). - ISBN 0-387-57705-X (New York: acid-free) 1. Approximation theory. 2. Polynomials. I. Title. II. Series: Lecture notes in mathematics (Springer-Verlag); 1569. QA3.L28 no. 1569 [QA221] 510 sdc20 [511'.42] 93-49416 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. 9 Springer-Verlag Berlin Heidelberg 1994 Printed in Germany SPIN: 10078788
46/3140-543210 - Printed on acid-free paper
Contents 1
I
Introduction
Freud weights
7
2
Short proof for the approximation
problem for Freud weights
3
Strong asymptotics 3.1 T h e upper estimate . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 The lower estimate . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 T h e / 2 case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7 10 14 17 19
II
Approximation with general weights
21
4
A general approximation theorem 4.1 Statement of the main results . . . . . . . . . . . . . . . . . . . . 4.2 Examples and historical notes . . . . . . . . . . . . . . . . . . . .
21 21 23
5
Preliminaries to the proofs
25
6
P r o o f o f T h e o r e m s 4.1, 4.2 a n d 4.3
32
7
C o n s t r u c t i o n o f E x a m p l e s 4.5 a n d 4.6 7.1 Example 4.5 . . . . . . . . . . . . . . . 7.2 Example 4.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
III 8
9
Varying weights Uniform approximation weights
by weighted polynomials with varying
in g e o m e t r i c m e a n s
Applications
by
49 57 59 60 64 7O
7'9
11 F a s t d e c r e a s i n g p o l y n o m i a l s 12 A p p r o x i m a t i o n
38 38 44
49
Modification of the method The lower estimate . . . . . . . . . . . . . . . . . . . . . . . . . . The upper estimate . . . . . . . . . . . . . . .
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