Approximation Theory Using Positive Linear Operators

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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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.

• Area
• Smooth function
• addition
• approximation
• approximation theory
• computer
• constant
• design
• field
• form
• function
• functional
• functional analysis
• functions
• types

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)

• Department of Mathematics, Transilvania University, Braşov, Romania

  Radu Păltănea

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