Shape-Preserving Approximation by Real and Complex Polynomials

ISBN-13: 9780817647025 / Angielski / Twarda / 2008 / 352 str.

Sorin G. Gal

Opis

In many problems arising in engineering and science one requires approxi- tion methods to reproduce physical reality as well as possible. Very schema- cally, if the input data represents a complicated discrete/continuous quantity of information, of shape S (S could mean, for example, that we have a monotone/convex collection of data), then one desires to represent it by the less-complicated output information, that approximates well the input data and, in addition, has the same shape S. This kind of approximation is called shape-preserving approximation and arises in computer-aided geometric design, robotics, chemistry, etc. Typically, the input data is represented by a real or complex function (of one or several variables), and the output data is chosen to be in one of the classes polynomial, spline, or rational functions. The present monograph deals in Chapters 1 4 with shape-preserving - proximation by real or complex polynomials in one or several variables. Chapter 5 is an exception and is devoted to some related important but n- polynomial andnonsplineapproximations preservingshape.Thesplinecaseis completely excluded in the present book, since on the one hand, many details concerning shape-preserving properties of splines can be found, for example, in the books of de Boor 49], Schumaker 344], Chui 69], DeVore Lorentz 91], Kvasov 218] and in the surveys of Leviatan 229], Koci c Milovanovi c 196], while on the other hand, we consider that shape-preserving approximation by splines deserves a complete study in a separate book."

Recenzje

From the reviews:

"This monograph is a rich collection of existing and new results in an important part of approximation theory, in which the object of approximation and the instrument for approximation share the same shape characteristics. ... This is convenient for a reference book ... . This book is a result of hard work and deep knowledge of the subject. It is a very important reference book for experts in approximation theory and would be a worthy addition to any mathematics library." (Bl. Sendov, Mathematical Reviews, Issue 2009 m)

Spis treści

Shape-Preserving Approximation By Real Univariate Polynomials.- Shape-Preserving Approximation by Real Multivariate Polynomials.- Shape-Preserving Approximation by Complex Univariate Polynomials.- Shape-Preserving Approximation by Complex Multivariate Polynomials.- Appendix : Some Related Topics.

Blurb

This monograph presents the first comprehensive treatment in book form of shape-preserving approximation by real or complex polynomials in one or several variables. Such approximation methods are useful in many problems that arise in science and engineering and require an optimal mathematical representation of physical reality.

The main topics are structured in four chapters, followed by an appendix: shape-preserving approximation and interpolation of real functions of one real variable by real polynomials; shape-preserving approximation of real functions of several real variables by multivariate real polynomials; shape-preserving approximation of analytic functions of one complex variable by complex polynomials in the unit disk; and shape-preserving approximation of analytic functions of several complex variables on the unit ball or the unit polydisk by polynomials of several complex variables. The appendix treats related results of non-polynomial and non-spline approximations preserving shape including those by complexified operators with applications to complex partial differential equations.

Shape-Preserving Approximation by Real and Complex Polynomials contains many open problems at the end of each chapter to stimulate future research along with a rich and updated bibliography surveying the vast literature. The text will be useful to graduate students and researchers interested in approximation theory, mathematical analysis, numerical analysis, computer aided geometric design, robotics, data fitting, chemistry, fluid mechanics, and engineering.