Approximation Theory: Moduli of Continuity and Global Smoothness Preservation - Hardcover

Anastassiou, George A.; Gal, Sorin G.

Hardcover

ISBN 10: 0817641513 ISBN 13: 9780817641511

Publisher: Birkhäuser, 1999

Synopsis

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.

About this title

Review

"This monograph is an intensive and comprehensive study of the computational aspects of the moduli of smoothness and the Global Smoothness Preservation Property (GSPP)."

---Zentralblatt MATH

"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 Math.

Synopsis

The key features of this text include: a study of the computation of moduli of continuity and GSPP; motivation and examples for key results; extensive applications of moduli of smoothness and GSPP concepts to approximation theory; probability theory; and numerical and functional analysis.

Edition details

• Publisher: Birkhäuser
• Publication date: 1999
• Language: English
• ISBN 10: 0817641513
• ISBN 13: 9780817641511
• Binding: Hardcover
• Number of pages: 538