The Approximation of Continuous Functions by Positive Linear Operators
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293 Ronald A. DeVore Oakland University, Rochester, MI/USA
The Approximation of Continuous Functions by Positive Linear Operators
Springer-Verlag Berlin-Heidelberg • NewYork 1972
AMS Subject Classifications (1970): 41-02, 41 A 10, 41 A25, 41A 35, 42 A04, 42A08
ISBN 3-540-06038-3 Springer-Verlag Berlin • Heidelberg • N e w York ISBN 0-387-06038-3 Springer-Verlag N e w York - Heidelberg • Berlin This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher. © by Springer-Vertag Berlin - Heidelberg 1972. Library of Congress Catalog Card Number 72-91891. Printed in Germany. Offsetdruck: Julius Beltz, Hemsbach/Bergstr.
PREFACE
These notes study linear methods of approximation which are given by a sequence
(Ln)
of positive linear operators.
that of positivlty. f
The essential ingredient being
The main theme is to relate the smoothness of the function
being approximated with the rate of decrease of
llf - en(f) ll .
This is
accomplished in the usual setting of direct theorems, inverse theorems, saturation and approximation of classes of functions. The fundamental ideas involved in direct estimates can be found in the pioneering book of P.P. K o r o v k i ~ and several of the more recent textbooks on approximation. first time.
However, most of the material appears in "book form" here for the
The main exception being the results on approximation by positive
convolution operators, which have considerable overlap with the recent book of P.L. Butzer and R.J. Nessel I. I have written the notes at a level which presupposes a knowledge of the fundamental aspects of approximation theory, especially as pertains to the degree of approximation.
Most of the necessary background material can be found in the
now classic book of G.G. Lorentz 2.
For a good understanding of the material
developed here for convolution operators, I expect that the reader will have to make several excursions into Butzer and Nessell. The notes concentrate solely on spaces of continuous function (periodic and non-periodic) on a finite interval.
I have not developed the theory for
L
P
-
spaces since I know of little in these spaces that goes beyond what is already contained in Butzer and Nessel I.
The only examples considered are those which fall
comfortably in the grasps of the general theory.
However, I believe that the
reader will find that most of the better known examples are covered.
ACKNOWLEDGEMENT.
I would like to thank Professor P.L. Butzer and his group at
Aachen, most notably Professor R.J. Nessel and Dr. 's E. G$°rlieh,E. Stark,K.Scherer, who were generous enough to cormnent on a rough first draft of these
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