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-Fejer 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 Stadtmuller. - 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 toGBPs2. - 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 Bernstein-Kantorovich Operators. - 19.4 Global Smoothness Preservation for Generalized Szasz-Kantorovich Operators. - 19.5 First Order Optimal Global Smoothness Preservation for Bernstein-Type Operators. - 20 Other Applications of the Global Smoothness Preservation Property. - 20.1 Relationships of the Global Smoothness Preservation Property with the Shape Preservation and the Variation Diminishing Properties. - 20.2 Global Smoothness Preservation in CAGD. - 20.3 Other Applications. - 20.4 Bibliographical Remarks. - References. - List of Symbols.