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 valยญ 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 Propยญ erty (GSPP) for almost all known linear approximation operators of apยญ proximation theory including: trigonometric operators and algebraic inยญ terpolation operators of Lagrange, Hermite-Fejer and Shepard type, also operators of stochastic type, convolution type, wavelet type integral operaยญ 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 mathematยญ ics such as Functional analysis, and outside of mathematics, fields such as computer-aided geometric design (CAGD). Most of the time GSPP methยญ 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
CONTENT
1 Introduction -- 1.1 On Chapter 2: Uniform Moduli of Smoothness -- 1.2 On Chapter 3: LP-Moduli of Smoothness, 1 ?p Trigonometric Operators -- 6 Global Smoothness Preservation by Algebraic Interpolation Operators -- 7 Global Smoothness Preservation by General Operators -- 8 Global Smoothness Preservation by Multivariate Operators -- 9 Stochastic Global Smoothness Preservation -- 10 Shift Invariant Univariate Integral Operators -- 11 Shift Invariant Multivariate Integral Operators -- 12 Differentiated Shift Invariant Univariate Integral Operators -- 13 Differentiated Shift Invariant Multivariate Integral Operators -- 14 Generalized Shift Invariant Univariate Integral Operators -- 15 Generalized Shift Invariant Multivariate Integral Operators -- 16 General Theory of Global Smoothness Preservation by Univariate Singular Operators -- 17 General Theory of Global Smoothness Preservation by Multivariate Singular Operators -- 18 Gonska Progress in Global Smoothness Preservation -- 19 Miscellaneous Progress in Global Smoothness Preservation -- 20 Other Applications of the Global Smoothness Preservation Property -- References -- List of Symbols
