Overview

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.

Full Product Details

Author:   George A. Anastassiou ,  Sorin G. Gal
Publisher:   Birkhauser Verlag AG
Imprint:   Birkhauser Verlag AG
ISBN:  

9783764341510

ISBN 10:   3764341513
Pages:   560
Publication Date:   March 2000
Audience:   College/higher education ,  Professional and scholarly ,  Undergraduate ,  Postgraduate, Research & Scholarly
Replaced By:   9780817641511
Format:   Hardback
Publisher's Status:   Active
Availability:   In Print  
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Table of Contents

Part 1 Calculus of the moduli of smoothness in classes of functions: uniform moduli of smoothness; LP-moduli of smoothness; moduli of smoothness of special type. Part 2 Global smoothness preservation by linear operators: global smoothness preservation by trigonometric operators; global smoothness preservation by algebraic interpolation operators; global smoothness preservation by general operators; global smoothness preservation by multivariate operators; stochastic global smoothness preservation; shift invariant univariate integral operators; shift invariant multivariate integral operators; differentiated shift invariant univariate integral operators; differentiated shift invariant multivariate integral operators; generalised shift invariant univariate integral operators; generalised shift invariant multivariate integral operators; general theory of global smoothness preservation by univariate singular integrals; general theory of global smoothness preservation by multivariate singular integrals; Ginska progress in global smoothness preservation; miscellaneous progress on global smoothness preservation; other applications of the global smoothness preservation property. References; list of symbols.

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