Ebook: Approximation Theory: Moduli of Continuity and Global Smoothness Preservation
- Tags: Applications of Mathematics, Approximations and Expansions, Global Analysis and Analysis on Manifolds, Analysis, Computational Mathematics and Numerical Analysis
- Year: 2000
- Publisher: Birkhäuser Basel
- Edition: 1
- Language: English
- pdf
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.
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). Key features include: * systematic and extensive study of the computation of Moduli of Continuity and GSPP, presented for the first time in book form * substantial motivation and examples for key results * extensive applications of moduli of smoothness and GSPP concepts to approximation theory, probability theory, numerical and functional analysis * GSPP methods to benefit engineers in computer-aided geometric design * good bibliography and index For researchers and graduate students in pure and applied mathematics.
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). Key features include: * systematic and extensive study of the computation of Moduli of Continuity and GSPP, presented for the first time in book form * substantial motivation and examples for key results * extensive applications of moduli of smoothness and GSPP concepts to approximation theory, probability theory, numerical and functional analysis * GSPP methods to benefit engineers in computer-aided geometric design * good bibliography and index For researchers and graduate students in pure and applied mathematics.
Content:
Front Matter....Pages i-xi
Introduction....Pages 1-53
Front Matter....Pages 55-55
Uniform Moduli of Smoothness....Pages 57-144
LP-Moduli of Smoothness,1 ?P <+?....Pages 145-169
Moduli of Smoothness of Special Type....Pages 171-199
Front Matter....Pages 201-201
Global Smoothness Preservation by Trigonometric Operators....Pages 203-210
Global Smoothness Preservation by Algebraic Interpolation Operators....Pages 211-230
Global Smoothness Preservation by General Operators....Pages 231-249
Global Smoothness Preservation by Multivariate Operators....Pages 251-263
Stochastic Global Smoothness Preservation....Pages 266-278
Shift Invariant Univariate Integral Operators....Pages 279-295
Shift Invariant Multivariate Integral Operators....Pages 297-323
Differentiated Shift Invariant Univariate Integral Operators....Pages 325-345
Differentiated Shift Invariant Multivariate Integral Operators....Pages 347-372
Generalized Shift Invariant Univariate Integral Operators....Pages 373-389
Generalized Shift Invariant Multivariate Integral Operators....Pages 391-400
General Theory of Global Smoothness Preservation by Univariate Singular Operators....Pages 401-427
General Theory of Global Smoothness Preservation by Multivariate Singular Operators....Pages 429-450
Gonska Progress in Global Smoothness Preservation....Pages 451-471
Miscellaneous Progress in Global Smoothness Preservation....Pages 473-484
Other Applications of the Global Smoothness Preservation Property....Pages 485-497
Back Matter....Pages 499-525
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). Key features include: * systematic and extensive study of the computation of Moduli of Continuity and GSPP, presented for the first time in book form * substantial motivation and examples for key results * extensive applications of moduli of smoothness and GSPP concepts to approximation theory, probability theory, numerical and functional analysis * GSPP methods to benefit engineers in computer-aided geometric design * good bibliography and index For researchers and graduate students in pure and applied mathematics.
Content:
Front Matter....Pages i-xi
Introduction....Pages 1-53
Front Matter....Pages 55-55
Uniform Moduli of Smoothness....Pages 57-144
LP-Moduli of Smoothness,1 ?P <+?....Pages 145-169
Moduli of Smoothness of Special Type....Pages 171-199
Front Matter....Pages 201-201
Global Smoothness Preservation by Trigonometric Operators....Pages 203-210
Global Smoothness Preservation by Algebraic Interpolation Operators....Pages 211-230
Global Smoothness Preservation by General Operators....Pages 231-249
Global Smoothness Preservation by Multivariate Operators....Pages 251-263
Stochastic Global Smoothness Preservation....Pages 266-278
Shift Invariant Univariate Integral Operators....Pages 279-295
Shift Invariant Multivariate Integral Operators....Pages 297-323
Differentiated Shift Invariant Univariate Integral Operators....Pages 325-345
Differentiated Shift Invariant Multivariate Integral Operators....Pages 347-372
Generalized Shift Invariant Univariate Integral Operators....Pages 373-389
Generalized Shift Invariant Multivariate Integral Operators....Pages 391-400
General Theory of Global Smoothness Preservation by Univariate Singular Operators....Pages 401-427
General Theory of Global Smoothness Preservation by Multivariate Singular Operators....Pages 429-450
Gonska Progress in Global Smoothness Preservation....Pages 451-471
Miscellaneous Progress in Global Smoothness Preservation....Pages 473-484
Other Applications of the Global Smoothness Preservation Property....Pages 485-497
Back Matter....Pages 499-525
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