Approximation Theory Using Positive Linear Operators

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.