Approximation Theory Using Positive Linear Operators

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.

Table of Contents:
1 Introduction.- 1.1 Operators and functionals. Moduli of continuity.- 1.2 Approximation of functions by sequences of positive linear operators.- 2 Estimates with Second Order Moduli.- 2.1 A general approach.- 2.2 Estimates with moduli ?2? and ?2?.- 2.3 Estimates with modulus ?2d.- 2.4 Estimates with modulus ?2dd.- 2.5 Estimates with Ditzian—Totik modulus.- 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?.- 4 Estimates for the Bernstein Operators.- 4.1 Various types of estimates.- 4.2 Best constant in the estimate with modulus ?2.- 4.3 Global smoothness preservation.- 5 Two Classes of Bernstein Type Operators.- 5.1 Generalized Brass type operators.- 5.2 Generalized Durrmeyer type operators.- 6 Approximation Operators for Vector-Valued Functions.- 6.1 Approximation of functions with real argument.- 6.2 Approximation of functions with vector argument.- References.

Review :
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)