Approximation Theory Using Positive Linear Operators - Radu Paltanea

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Résumé :
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. New and efficient methods, applicable to Bernstein operators and to diverse concrete moduli, are presented in this book. The advantages of these methods consist in obtaining improved and even optimal estimates, as well as in broadening the applicability of the 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....

Sommaire:
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 dollars dollars {{\tilde{\omega }}_{2}} dollars dollars.- References.