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Approximation Theory Using Positive Linear Operators

  • Book
  • © 2004

Overview

Authors:
  1. Radu Păltănea

    1. Department of Mathematics, Transilvania University, Braşov, Romania

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  • Approximation theory using positive linear operators has been a key area of research in the last few decades. Focus on the current important topic of Bernstein operators with excellent new applications to 2 new classes of Bernstein operators
  • New and efficient methods, which can produce improved and even optimal estimates, as well as broaden the applicability of the results
  • Many general estimates, leaving room for future applications

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Table of contents (6 chapters)

  1. Front Matter

    Pages i-ix
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  2. Introduction

    • Radu Păltănea
    Pages 1-14

  3. Estimates with Second Order Moduli

    • Radu Păltănea
    Pages 15-68

  4. Absolute Optimal Constants

    • Radu Păltănea
    Pages 69-87

  5. Estimates for the Bernstein Operators

    • Radu Păltănea
    Pages 89-129

  6. Two Classes of Bernstein Type Operators

    • Radu Păltănea
    Pages 131-159

  7. Approximation Operators for Vector-Valued Functions

    • Radu Păltănea
    Pages 161-193

  8. Back Matter

    Pages 195-202
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Keywords

About this book

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.

Reviews

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)

Authors and Affiliations

  • Department of Mathematics, Transilvania University, Braşov, Romania

    Radu Păltănea

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