Approximation Theory: Moduli of Continuity and Global Smoothness Preservation

1 Introduction.- 1.1 On Chapter 2: Uniform Moduli of Smoothness.- 1.2 On Chapter 3: LP-Moduli of Smoothness, 1 ?p Modulus of Smoothness.- 2.5 Applications.- 2.6 Bibliographical Remarks and Open Problems.- 3 LP-Moduli of Smoothness, 1 ?p +?.- 3.1 Usual LP -Modulus of Smoothness.- 3.2 Averaged LP -Modulus of Smoothness.- 3.3 Ditzian-Totik LP -Modulus of Smoothness.- 3.4 Applications.- 3.5 Bibliographical Remarks and Open Problems.- 4 Moduli of Smoothness of Special Type.- 4.1 One-Sided Modulus of Smoothness.- 4.2 Hausdorff-Sendov Modulus of Continuity.- 4.3 An Algebraic Modulus of Smoothness.- 4.4 Weighted Moduli of Smoothness.- 4.5 Applications.- 4.6 Bibliographical Remarks and Open Problems.- II Global Smoothness Preservation by Linear Operators.- 5 Global Smoothness Preservation by Trigonometric Operators.- 5.1 General Results.- 5.2 Global Smoothness Preservation by Some Concrete Trigonometric Operators.- 5.3 Global Smoothness Preservation by Trigonometric Projection Operators.- 5.4 Bibliographical Remarks and Open Problems.- 6 Global Smoothness Preservation by Algebraic Interpolation Operators.- 6.1 Negative Results.- 6.2 Global Smoothness Preservation by Some Lagrange, Hermite-Fejér and Shepard Operators.- 6.3 Global Smoothness Preservation by Algebraic Projection Operators.- 6.4 Global Smoothness Preservation by Algebraic Polynomials of Best Approximation.- 6.5 Bibliographical Remarks and Open Problems.- 7 Global Smoothness Preservation by General Operators.- 7.1 Introduction.- 7.2 General Results.- 7.3 Applications.- 7.3.1 Variation-Diminishing Splines.- 7.3.2 Operators of Kratz and Stadtmüller.- 7.4 Optimality of the Preceding Results.- 8 Global Smoothness Preservation by Multivariate Operators.- 8.1 Introduction.- 8.2 A General Result for Operators Possessing the Splitting Property.- 8.3 Bernstein Operators over Simplices.- 8.4 Tensor Product Bernstein Operators.- 8.5 An Identity Between K-Functionals and More Results on Global Smoothness.- 8.6 Example: A Comparison Theorem in Stochastic Approximation.- 9 Stochastic Global Smoothness Preservation.- 9.1 Introduction.- 9.2 Preliminaries.- 9.3 A Theorem on Stochastic Global Smoothness Preservation.- 9.4 Applications.- 9.4.1 Stochastic Convolution-Type Operators on C?0 [a, b].- 9.4.2 Operators on C?[[a, b].- 9.4.3 More Convolution-Type Operators.- 10 Shift Invariant Univariate Integral Operators.- 10.1 Introduction.- 10.2 General Theory.- 10.3 Applications.- 11 Shift Invariant Multivariate Integral Operators.- 11.1 General Results.- 11.2 Applications.- 12 Differentiated Shift Invariant Univariate Integral Operators.- 12.1 Introduction.- 12.1.1 Other Motivations.- 12.2 General Results.- 12.3 Applications.- 13 Differentiated Shift Invariant Multivariate Integral Operators.- 13.1 Introduction.- 13.2 General Results.- 13.3 Applications.- 14 Generalized Shift Invariant Univariate Integral Operators.- 14.1 General Theory.- 14.2 Applications.- 15 Generalized Shift Invariant Multivariate Integral Operators.- 15.1 General Theory.- 15.2 Applications.- 16 General Theory of Global Smoothness Preservation by Univariate Singular Operators.- 16.1 Introduction.- 16.2 General Theory.- 17 General Theory of Global Smoothness Preservation by Multivariate Singular Operators.- 17.1 Introduction.- 17.2 General Results.- 18 Gonska Progress in Global Smoothness Preservation.- 18.1 Simultaneous Global Smoothness Preservation.- 18.2 Bivariate Global Smoothness Preservation by Boolean Sum Operators.- 18.3 Global Smoothness Preservation with Respect to Pds. s2.- 18.4 Global Smoothness Preservation for Bernstein Polynomials Blossoms.- 18.5 Global Smoothness Preservation for Boolean Sums of Convolution Type Operators.- 19 Miscellaneous Progress in Global Smoothness Preservation.- 19.1 Preservation of Lipschitz Classes by Bernstein-Type Operators.- 19.2 Preservation of Lipschitz Classes by Some Positive Linear Operators over Unbounded Intervals.- 19.3 Global Smoothness Preservation of Generalized Ber