Approximation theory using positive linear operators
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Bibliographic Information
Approximation theory using positive linear operators
Birkhäuser, c2004
Available at / 8 libraries
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Hokkaido University, Library, Graduate School of Science, Faculty of Science and School of Science図書
DC22:511.4/P1882080022942
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Includes bibliographical references and index
Description and Table of Contents
Description
Offers an 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
Table of Contents
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 $$ {<!-- -->{\tilde{\omega }}_{2}} $$.- References.
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