Approximation Theory Using Positive Linear Operators

207 Pages·2004·6.34 MB·English

by  Radu Păltănea (auth.)

Consulting Editor
George A. Anastassiou
Department of Mathematical Sciences
University of Memphis
Radu PiHtanea
Approximation Theory
Us ing Positive
Linear Operators
Birkhauser
Boston • Basel • Berlin
Radu PAltmlea
Transilvania University
Department of Mathematics
Bra§ov 500 036
Romania
AMS Subject Classifications: Primary: 41A36; Secondary: 41A35, 41A25, 41A44, 41AlO, 41A28, 41A65,
41A46, 26B25, 32A70, 46A32, 47Axx, 47A67, 47A63, 47B38, 47A65, 39B62, 26A15, 26A51, 26B25,
26D15, 28Axx, 05AI9, 05A20
Library of Congress Cataloging-in-PubHcation Data
Paltanea, Radu, 1953-
Approximation theory using positive linear operators I Radu Paltanea.
p.cm.
Includes bibliographical references and index.
ISBN-13:978-0-8176-4350-8 e-ISBN-13:978-1-4612-2058-9
DOl: 10.10071978-1-4612-2058-9
I. Approximation theory. 2. Linear operators. I. Title.
QA221.P25 2004
51 I '.4-dc22 2004054852
ISBN -13 :978-0-8176-4350-8 Printed on acid-free paper.
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Preface
We deal in this work with quantitative results in the pointwise approximation of func
tions by positive linear functionals and operators.
One of the main objectives is to obtain estimates for the degree of approximation
in terms of various types of second order moduli of continuity. In the category of sec
ond order moduli we include both classical and newly introduced moduli. Particular
attention is paid to optimizing the constants appearing in such estimates.
In the last decades, the study of linear positive operators with the aid of second
order moduli was intensive, thanks to their refinements in characterization of the
smoothness of functions. As promoters of this direction of research we mention Yu.
Brudnyi, G. Freud, and J. Petree. Our approach is more akin to the approach taken
by H. Gonska, who obtained the first general estimates for second order moduli with
precise constants and with free parameters. Two new methods will be presented. The
first one, based on decomposition of functionals and the use of moments, can be
applied to diverse types of moduli and leads to simple estimates. The second method
gives sufficient conditions for obtaining absolute optimal constants. The benefits of
these more direct methods, compared with the known method based on K-functionals,
consist in the improvement and even the optimization of the constants, and in the
generalization of the framework.
Applications are given for the classical Bernstein operators and for two classes of
Bernstein-type operators. So, for the Bernstein operators, we prove the analogous op
timality result of Sikkema for the second order modulus. The first class of Bernstein
type operators on which we focus our attention is a class of certain generalized Brass
operators that admit a construction based on discrete convolution. The second class
consists of Durrmeyer integral operators with generalized weights. The Durrmeyer
operators, introduced also by A. Lupa§, were intensively studied by M. Dierrennic
and many other authors.
In the last part we consider the problem of approximation of vector functions by
using certain generalizations of the positive and convex operators. The quantitative
results are similar to those given in the scalar case.
vi Preface
Acknowledgments. The author is deepley indebted to Professor Heinz H. Gonska
and to Professor George Anastassiou, for their support and for promoting the work.
Also we are grateful to the Birkhauser publishing house for accepting the publication
of the book and also to Ann Kostant, Regina Gorenshteyn and Elizabeth Loew for
their kind assistance.
Radu Paltanea
Contents
Preface......................................................... v
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Operators and functionals. Moduli of continuity. . . . . . . . . . . . . . . . . . . 1
1.2 Approximation of functions by sequences of positive linear operators. 7
1.2.1 Basic theorems of convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.2.2 Estimates with the first order modulus . . . . . . . . . . . . . . . . . . . . . 9
2 Estimates with Second Order Moduli. . ... . . . . .. . . . . . . . . . . . . . . . . . 15
2.1 A general approach.. . . . .. . . . . ... . . .... . ... .. .. . . .. . . . . . . . . ... 15
2.1.1 Introduction........................................... 15
2.1.2 A general estimate for the degree of approximation by linear
positive functionals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 16
2.2 Estimates with moduli w~ and wi... . . ... . . ... . ..... .. ...... .. .. 21
2.2.1 Introduction. Auxiliary results. . . . . . . . . . . . . . . . . . . . . . . . . . .. 21
2.2.2 Main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 27
2.3 Estimates with modulus w~ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 34
2.3.1 Introduction. Auxiliary results. . . . . . . . . . . . . . . . . . . . . . . . . . .. 34
2.3.2 Main results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 38
2.4 Estimates with modulus w~d ................................... 50
2.5 Estimates with Ditzian-Totik modulus. . . . . . . . . . . . . . . . . . . . . . . . . .. 54
2.5.1 Auxiliary results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 55
2.5.2 Main result. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 64
3 Absolute Optimal Constants ................................... 69
3.1 Introduction................................................. 69
3.2 Discrete functionals and the classical second order modulus W2 . . . . .. 72
3.3 General functionals and the second order modulus with parameter w~. 76
3.3.1 A particular case. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 76
3.3.2 The main results ....................................... 81
viii Contents
4 Estimates for the Bernstein Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
4.1 Various types of estimates ..................................... 89
4.1.1 Introduction........................................... 89
4.1.2 Applications of general estimates ......................... 93
4.2 Best constant in the estimate with modulus CO2 .................... 96
4.2.1 Introduction. Main result .......................... . . . . .. 96
4.2.2 Proof of the direct part of the theorem for n ::: 60 . . . . . . . . . . .. 98
4.2.3 Proof of the direct part of the theorem for 1 :::: n :::: 59 ........ 110
4.3 Global smoothness preservation ................................ 125
5 Two Classes of Bernstein Type Operators ........................ 131
5.1 Generalized Brass type operators ............................... 131
5.1.1 Definitions and general properties ......................... 131
5.1.2 Simultaneous approximation ............................. 139
5.2 Generalized Durrmeyer type operators ........................... 146
5.2.1 Durrmeyer type operators with general weights ............. 146
5.2.2 Durrmeyer type operators with generalized Jacobi weights .... 151
6 Approximation Operators for Vector-Valued Functions ............. 161
6.1 Approximation of functions with real argument ................... 161
6.1.1 Introduction. Generalized positive and convex operators ...... 161
6.1.2 A Korovkin type theorem ................................ 162
6.1.3 Simultaneous approximation ............................. 165
6.2 Approximation of functions with vector argument ................. 171
6.2.1 Introduction. Linear functionals and operators induced by
positive measures ...................................... 171
6.2.2 Auxilliary results ....................................... 175
wi .........................
6.2.3 Estimates with moduli CO2 and 179
6.2.4 Estimates with modulus W2 .............................. 187
References .... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 195
Index ........................................................... 201
Approximation Theory
Using Positive
Linear Operators
1
Introduction
1.1 Operators and functionals. Moduli of continuity
The most constructive proofs of the Weierstrass theorem concerning the approxi
mation of continuous functions on a compact interval by polynomials use some
sequences of linear positive operators. So do the classical operators of Gauss
Weierstrass, Landau, Vallee-Poussin, Jackson, and Bernstein. We begin by construct
ing and studying a large class of such sequences of approximation operators.
We mention that the more usual tools of Analysis, like the Taylor polynomials,
the Fourier series related to an inner product, or the Lagrange interpolations, are not
appropriate to the problem of the uniform approximation of continuous functions.
An operator L defined on a linear space of functions, V, is called linear if
L(af + fJg) = aL(f) + fJL(g), for all f, g E V, a, fJ E lR, (1.1)
and is called positive, if
L(f) ::: 0, for all f E V, f::: O. (1.2)
These properties are not necessary in order for a sequence of operators to give
a uniform approximation to any continuous function. So, the sequence of the oper
ators that assign to each continuous function its best polynomial approximation, of
degree n, with respect to the sup-norm is not linear and is not positive. But these
properties simplify study to a considerable degree. On the other hand, the value of
the approximation may, though reduced somewhat in degree of goodness, consist of
some other qualities. We mention for instance the preservation of some properties
of the functions, the possibility of approximation of derivatives, or the possibility of
characterization of certain classes of functions by the order of approximation that can
be reached.
We specify now some notation that will be used in the whole paper. Particular
notation will be introduced along the way.
Denote N := {l, 2, ... }. For n E Nu {OJ, fln is the space of algebric polynomials
of degree at most n. For any real number a, denote by [a] the integer part of a, by ]a [
R. Păltănea, Approximation Theory Using Positive Linear Operators
© Birkhäuser Boston 2004