Approximation Theory Using Positive Linear Operators

ISBN-13: 9780817643508 / Angielski / Miękka / 2004 / 216 str.

Radu Paltanea

Opis

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."

Recenzje

From the reviews:

"This monograph will be of interest to those working in the field of approximation, and it may serve as a good supplementary text for courses in approximation theory, or as a reference text on the subject."(ZENTRALBLATT MATH)

Spis treści

1 Introduction.- 1.1 Operators and functionals. Moduli of continuity.- 1.2 Approximation of functions by sequences of positive linear operators.- 1.2.1 Basic theorems of convergence.- 1.2.2 Estimates with the first order modulus.- 2 Estimates with Second Order Moduli.- 2.1 A general approach.- 2.1.1 Introduction.- 2.1.2 A general estimate for the degree of approximation by linear positive functionals.- 2.2 Estimates with moduli ?2? and ?2?.- 2.2.1 Introduction. Auxiliary results.- 2.2.2 Main results.- 2.3 Estimates with modulus ?2d.- 2.3.1 Introduction. Auxiliary results.- 2.3.2 Main results.- 2.4 Estimates with modulus ?2dd.- 2.5 Estimates with Ditzian—Totik modulus.- 2.5.1 Auxiliary results.- 2.5.2 Main result.- 3 Absolute Optimal Constants.- 3.1 Introduction.- 3.2 Discrete functionals and the classical second order modulus ?2.- 3.3 General functionals and the second order modulus with parameter ?2?.- 3.3.1 A particular case.- 3.3.2 The main results.- 4 Estimates for the Bernstein Operators.- 4.1 Various types of estimates.- 4.1.1 Introduction.- 4.1.2 Applications of general estimates.- 4.2 Best constant in the estimate with modulus ?2.- 4.2.1 Introduction. Main result.- 4.2.2 Proof of the direct part of the theorem for n ? 60.- 4.2.3 Proof of the direct part of the theorem for 1 ? n ? 59.- 4.3 Global smoothness preservation.- 5 Two Classes of Bernstein Type Operators.- 5.1 Generalized Brass type operators.- 5.1.1 Definitions and general properties.- 5.1.2 Simultaneous approximation.- 5.2 Generalized Durrmeyer type operators.- 5.2.1 Durrmeyer type operators with general weights.- 5.2.2 Durrmeyer type operators with generalized Jacobi weights.- 6 Approximation Operators for Vector-Valued Functions.- 6.1 Approximation of functions with real argument.- 6.1.1 Introduction. Generalized positive and convex operators.- 6.1.2 A Korovkin type theorem.- 6.1.3 Simultaneous approximation.- 6.2 Approximation of functions with vector argument.- 6.2.1 Introduction. Linear functionals and operators induced by positive measures.- 6.2.2 Auxilliary results.- 6.2.3 Estimates with moduli ?2 and ?2?.- 6.2.4 Estimates with modulus $$ {{\tilde{\omega }}_} $$.- References.

Blurb

This work treats quantitative aspects of the approximation of functions using positive linear operators. The theory of these operators has been an important area of research in the last few decades, particularly as it affects computer-aided geometric design. In this book, the crucial role of the second order moduli of continuity in the study of such operators is emphasized. New and efficient methods, applicable to general operators and to diverse concrete moduli, are presented. The advantages of these methods consist in obtaining improved and even optimal estimates, as well as in broadening the applicability of the results.

Additional Topics and Features:

*  Examination of the multivariate approximation case

*  Special focus on the Bernstein operators, including applications, and on two new classes of Bernstein-type operators

*  Many general estimates, leaving room for future applications (e.g. the B-spline case)

*  Extensions to approximation operators acting on spaces of vector functions

*  Historical perspective in the form of previous significant results

This monograph will be of interest to those working in the field of approximation or functional analysis. Requiring only familiarity with the basics of approximation theory, the book may serve as a good supplementary text for courses in approximation theory, or as a reference text on the subject.