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£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 Bernstein-Kantorovich Operators.- 19.4 Global Smoothness Preservation for Generalized Szász-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.
We study in Part I of this monograph the computational aspect of almost all moduli of continuity over wide classes of functions exploiting some of their convexity properties. To our knowledge it is the first time the entire calculus of moduli of smoothness has been included in a book. We then present numerous applications of Approximation Theory, giving exact val ues of errors in explicit forms. The K-functional method is systematically avoided since it produces nonexplicit constants. All other related books so far have allocated very little space to the computational aspect of moduli of smoothness. In Part II, we study/examine the Global Smoothness Preservation Prop erty (GSPP) for almost all known linear approximation operators of ap proximation theory including: trigonometric operators and algebraic in terpolation operators of Lagrange, Hermite-Fejer and Shepard type, also operators of stochastic type, convolution type, wavelet type integral opera tors and singular integral operators, etc. We present also a sufficient general theory for GSPP to hold true. We provide a great variety of applications of GSPP to Approximation Theory and many other fields of mathemat ics such as Functional analysis, and outside of mathematics, fields such as computer-aided geometric design (CAGD). Most of the time GSPP meth ods are optimal. Various moduli of smoothness are intensively involved in Part II. Therefore, methods from Part I can be used to calculate exactly the error of global smoothness preservation. It is the first time in the literature that a book has studied GSPP.
"Going over the introduction, the reader will get an almost full account, without proofs, of the results presented in the monograph... There are many questions for further research arising from this interesting book. Some of them are formulated by the authors, but many more may be born in the mind of the reader." ---Zentralblatt
This monograph, in two parts, is an intensive and comprehensive study of the computational aspects of the moduli of smoothness and the Global Smoothness Preservation Property (GSPP).