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© 2000

Approximation Theory

Moduli of Continuity and Global Smoothness Preservation

Book

Table of contents

  1. Front Matter
    Pages i-xi
  2. Introduction

    1. George A. Anastassiou, Sorin G. Gal
      Pages 1-53
  3. Calculus of the Moduli of Smoothness in Classes of Functions

    1. Front Matter
      Pages 55-55
    2. George A. Anastassiou, Sorin G. Gal
      Pages 57-144
    3. George A. Anastassiou, Sorin G. Gal
      Pages 145-169
    4. George A. Anastassiou, Sorin G. Gal
      Pages 171-199
  4. Global Smoothness Preservation by Linear Operators

    1. Front Matter
      Pages 201-201
    2. George A. Anastassiou, Sorin G. Gal
      Pages 203-210
    3. George A. Anastassiou, Sorin G. Gal
      Pages 211-230
    4. George A. Anastassiou, Sorin G. Gal
      Pages 231-249
    5. George A. Anastassiou, Sorin G. Gal
      Pages 251-263
    6. George A. Anastassiou, Sorin G. Gal
      Pages 266-278
    7. George A. Anastassiou, Sorin G. Gal
      Pages 279-295
    8. George A. Anastassiou, Sorin G. Gal
      Pages 297-323
    9. George A. Anastassiou, Sorin G. Gal
      Pages 325-345
    10. George A. Anastassiou, Sorin G. Gal
      Pages 347-372
    11. George A. Anastassiou, Sorin G. Gal
      Pages 373-389
    12. George A. Anastassiou, Sorin G. Gal
      Pages 391-400

About this book

Introduction

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.

Keywords

Approximation Complex Analysis Interpolation Numerical Analysis Smooth function approximation theory functional analysis linear optimization pdc

Authors and affiliations

  1. 1.Department of Mathematical SciencesUniversity of MemphisMemphisUSA
  2. 2.Department of MathematicsUniversity of OradeaOradeaRomania

Bibliographic information

Reviews

"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.