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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.

Inhaltsverzeichnis

Frontmatter

Introduction

1. Introduction

Abstract
To convey some of the essence and following of this monograph to the reader, in this chapter we present briefly some of its main results as well as to include some important motivation for writing it. For the convenience of the reader, the exposed results are numbered as they are in their respective chapters.
George A. Anastassiou, Sorin G. Gal

Calculus of the Moduli of Smoothness in Classes of Functions

Frontmatter

2. Uniform Moduli of Smoothness

Abstract
In this chapter we obtain a few results concerning the calculus and estimate of the main known uniform moduli of smoothness for some subclasses of functions of one or several variables. Section 2.5 contains applications of these results and Section 2.6 contains bibliographical remarks and open problems.
George A. Anastassiou, Sorin G. Gal

3. LP-Moduli of Smoothness,1 ≤P <+∞

Abstract
This chapter discusses the calculus and estimates of the most used L P -moduli of smoothness, 1 ≤p <+∞, in some subclasses of functions.
George A. Anastassiou, Sorin G. Gal

4. Moduli of Smoothness of Special Type

Abstract
In this chapter we study from computational point of view some special types of moduli of smoothness, different from those in the Chapters 2 and 3 but they appear in many important cases in approximation theory.
George A. Anastassiou, Sorin G. Gal

Global Smoothness Preservation by Linear Operators

Frontmatter

5. Global Smoothness Preservation by Trigonometric Operators

Abstract
Let {Tn(f)}n be a sequence of trigonometric approximation operators applied to a continuous periodic function fC.
George A. Anastassiou, Sorin G. Gal

6. Global Smoothness Preservation by Algebraic Interpolation Operators

Abstract
Let {Ln(f)}n be a sequence of approximation algebraic operators, applied to a non-periodic function f ∈ C[a, b]. When {Ln(f)}n does not preserve the global smoothness of f,a natural question arises: how much of the global smoothness of f is preserved by {L n (f)} n ?
George A. Anastassiou, Sorin G. Gal

7. Global Smoothness Preservation by General Operators

Abstract
In this chapter we search the conditions under which global smoothness of a function f (as measured by its modulus of continuity) is preserved by the elements of general approximating sequences (L n f). As one consequence we obtain statements concerning the invariance of Lipschitz classes under operators of several types. An important tool in our approach is the least concave majorant of a modulus of continuity. Here we follow the basic study done by the first author, Cottin and Gonska [22].
George A. Anastassiou, Sorin G. Gal

8. Global Smoothness Preservation by Multivariate Operators

Abstract
In this chapter we discuss the global smoothness preservation by some multivariate approximating operators. By extending a fundamental result of Khan and Peters, we establish a general result for operators having the splitting property. Furthermore, we show more complete inequalities for Bernstein operators on the k-dimensional simplex and cube, formulate a certain transfer principle for tensor product operators, and apply an earlier related result in the context of stochastic approximation. Here we follow the basic study done by the first author, Cottin and Gonska [23].
George A. Anastassiou, Sorin G. Gal

9. Stochastic Global Smoothness Preservation

Abstract
Let (Ω, A,P) be a probability space and let CΩ[a, b]denote the space of stochastically continuous stochastic processes with index set [a,b]. When C [a,b] ⊂ VCΩ[a,b] and \( \tilde L:V \to C_\Omega \left[ {a,b} \right] \) is an E(expectation)-commutative linear operator on V, sufficient conditions are given here for E-preservation of global smoothness of XV through \( \tilde L \). Namely, it is given that
$$ {\omega _1}(E(\widetilde LX);\delta \leqslant \left\| L \right\|.{\widetilde \omega _1}\left( {EX;\frac{{c.\delta }}{{\left\| L \right\|}}} \right) \leqslant (\left\| L \right\| + c).{\omega _1}(EX;\delta ) $$
, where \( L: = \tilde L|_{C\left[ {a,b} \right]} \) , and for 0 ≤ δ ≤ b-a, ω 1 denotes the first order modulus of continuity with \( \tilde \omega _1 \) its least concave majorant and c a universal constant. Applications are given to different types of stochastic convolution operators defined through a kernel. Especially are studied extensively in this connection, stochastic operators defined through a bell-shaped trigonometric kernel. Another application of the above result is to stochastic discretely defined Kratz and Stadtmüller operators.
George A. Anastassiou, Sorin G. Gal

10. Shift Invariant Univariate Integral Operators

Abstract
In this chapter among other topics, we further study global smoothness over R. A very general positive linear integral type operator is given through a convolution-like iteration of another general positive linear operator with a scaling type function. For it sufficient conditions are given for shift invariance, preservation of global smoothness, convergence to the unit with rates, shape preserving and preservation of continuous probabilistic distribution functions. Furthermore, examples of very general specialized operators are presented fulfilling all the above properties; especially, the inequalities for global smoothness preservation are proven to be sharp. Here we follow the basic study done by the first author and Gonska [30].
George A. Anastassiou, Sorin G. Gal

11. Shift Invariant Multivariate Integral Operators

Abstract
Here among other topics we further study global smoothness preservation over Rd,d ≥ 1. This is a generalization of Chapter 10 in the multivariate case. Namely, a general positive linear multivariate integral type operator is given through a convolution-like iteration of another general positive linear multivariate operator with a multivariate scaling type function. For this sufficient conditions are given for shift invariance, global smoothness preservation and its sharpness, convergence to the unit with rates, shape preserving and preservation of continuous probabilistic distribution functions. Additionally, four examples of general specialized multivariate operators are given fulfilling all the above properties; especially, the inequalities for global smoothness preservation are sharp. In this chapter global smoothness preservation and convergence to the unit with rates involve a naturally arising suitable multivariate modulus of continuity. Here we follow the basic study done by the first author and Gonska [29]. Chapter 8 is also related here.
George A. Anastassiou, Sorin G. Gal

12. Differentiated Shift Invariant Univariate Integral Operators

Abstract
This is a continuation of Chapter 10 among others, still we study global smoothness preservation over R. Here are given sufficient conditions, so that the derivatives of general operators, examined in Chapter 10, enjoy the same nice properties as their originals. A sufficient condition is also given so that the “global smoothness preservation” related inequality becomes sharp. At the end of the chapter several applications are presented, where the derivatives of the very general specialized operators are shown to fulfill all the related properties. In particular it is established that they preserve continuous probability density functions. Here we follow the basic study done by the first author [21].
George A. Anastassiou, Sorin G. Gal

13. Differentiated Shift Invariant Multivariate Integral Operators

Abstract
This is a continuation of Chapter 11, among others, we still study global smoothness preservation over R d ,d > 1. Here are given sufficient conditions, so that the partial derivatives of general multivariate operators, examined in Chapter 11, enjoy most of the nice properties of their originals. Especially a sufficient condition is given so that the “global smoothness preservation” corresponding multivariate inequality is attained, that is sharp. Finally several applications are given, there the partial derivatives of very general specialized multivariate operators are shown to fulfill most of in Chapter 11 mentioned properties. In particular the partials of these operators are shown to preserve continuous multivariate probability density functions. Here we follow the basic study [20].
George A. Anastassiou, Sorin G. Gal

14. Generalized Shift Invariant Univariate Integral Operators

Abstract
This chapter is a continuation and generalization of Chapter 10. Among others we further study global smoothness preservation over R. In particular, certain others, similar to those in Chapter 10, but more general integral operators are presented and studied. These operators arise in a natural way. And for all these are given sufficient conditions for: shift invariance, preservation of higher order global smoothness and sharpness of the related inequalities, convergence to the unit using the first modulus of continuity, shape preserving and preservation of continuous probabilistic distribution functions. Several examples of diverse, very general specialized operators are given fulfilling all the above listed properties. Here we follow the basic study done by both authors in [24].
George A. Anastassiou, Sorin G. Gal

15. Generalized Shift Invariant Multivariate Integral Operators

Abstract
This chapter is a continuation and generalization of Chapters 11 and 14. Among others we further study global smoothness preservation over Rd,d ≥ 1. In particular, certain other similar to those in Chapter 11, but more general, multivariate integral operators are presented and studied. These operators come up naturally. And for all these are given sufficient conditions for multivariate: shift invariance, preservation of higher order global smoothness and sharpness of the related inequalities, convergence to the unit using the first modulus of continuity with respect to uniform norm, shape preserving on R d, and preservation of multivariate continuous probabilistic distribution functions. Several examples of diverse very general but specified multivariate integral operators fulfilling this theory are given at the end. Here we follow the basic study done by both authors in [25].
George A. Anastassiou, Sorin G. Gal

16. General Theory of Global Smoothness Preservation by Univariate Singular Operators

Abstract
In this chapter we show that the well-known singular integrals of Picard, Poisson-Cauchy, Gauss-Weierstrass and their Jackson type generalizations satisfy the “global smoothness preservation” property. I.e., they “ripple” less than the function they are applied on, that is producing a nice and fit approximation to the unit. The related results are given over various spaces of functions and the associated inequalities involve different types of corresponding moduli of smoothness. Several times these inequalities are proved to be sharp, namely they are attained. Here we follow the basic study done by both authors [26].
George A. Anastassiou, Sorin G. Gal

17. General Theory of Global Smoothness Preservation by Multivariate Singular Operators

Abstract
This is a continuation and generalization in the multivariate case of Chapter 16. Namely, by using various kinds of multivariate moduli of smoothness, in this chapter is presented that the multivariate variants of the well-known singular integrals of Picard, Poisson-Cauchy, Gauss-Weierstrass and their Jackson type generalizations satisfy the “global smoothness preservation” property. Here we follow the basic study done by both authors [27].
George A. Anastassiou, Sorin G. Gal

18. Gonska Progress in Global Smoothness Preservation

Abstract
Here global smoothness is mainly expressed by the Peetre K-functional of order s ≥1, defined by
$$ \begin{gathered} K_S \left( {f,\delta } \right): = K\left( {f;\delta ;C\left( {\left[ {0,1} \right]} \right),C^s \left( {\left[ {0,1} \right]} \right)} \right) \hfill \\ : = \inf \left\{ {\parallel f - g\parallel _\infty :g \in C^s \left( {\left[ {0,1} \right]} \right)} \right\} \hfill \\ \end{gathered} $$
where fC([0,1]),δ ≥0, and ∥ · ∥ is the supremum norm.
George A. Anastassiou, Sorin G. Gal

19. Miscellaneous Progress in Global Smoothness Preservation

Abstract
Here we follow/present the work of Della Vechia and Rasa [92].
George A. Anastassiou, Sorin G. Gal

20. Other Applications of the Global Smoothness Preservation Property

Abstract
In this last chapter we discuss some implications of the global smoothness preservation phenomenon in various fields of mathematics.
George A. Anastassiou, Sorin G. Gal

Backmatter

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