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Shape-Preserving Approximation by Real and Complex Polynomials - Sorin G. Gal

Shape-Preserving Approximation by Real and Complex Polynomials

Hardcover Published: 1st June 2008
ISBN: 9780817647025
Number Of Pages: 352

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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.

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)

Prefacep. vii
Shape-Preserving Approximation by Real Univariate Polynomialsp. 1
Introductionp. 1
Shape-Preserving Interpolation by Polynomialsp. 7
Bernstein-Type Polynomials Preserving Shapesp. 19
Shisha-Type Resultsp. 35
Positive and Copositive Polynomial Approximationp. 38
Pointwise Positive Approximationp. 38
L[superscript p]-Positive Approximation, 0 < p < [infinity]p. 39
Uniform and Pointwise Copositive Approximationp. 41
L[superscript p]-Copositive Approximation, 0 < p < [infinity]p. 47
Copositive Approximation with Modified Weighted Moduli of Smoothnessp. 48
Generalizationsp. 50
Monotone and Comonotone Polynomial Approximationp. 54
L[superscript p]-Monotone Approximation, 0 < p [less than or equal infinity]p. 57
Pointwise Monotone Approximationp. 62
L[superscript p]-Comonotone Approximation, 0 < p [less than or equal infinity]p. 64
Comonotone Approximation with Modified Weighted Moduli of Smoothnessp. 68
Nearly Comonotone Approximationp. 70
Convex and Coconvex Polynomial Approximationp. 73
Linear Methods in Convex Approximationp. 74
Nonlinear Methods in Convex Approximationp. 80
Pointwise Convex Approximationp. 81
Convex Approximation with Modified Weighted Moduli of Smoothnessp. 82
Uniform Coconvex Approximationp. 83
Coconvex Approximation with Modified Weighted Moduli of Smoothnessp. 85
Pointwise Coconvex Approximationp. 86
Nearly Coconvex Approximationp. 87
Shape-Preserving Approximation by Convolution Polynomialsp. 90
Positive Linear Polynomial Operators Preserving Shapep. 94
Notesp. 95
Shape-Preserving Approximation by Real Multivariate Polynomialsp. 99
Introductionp. 99
Bernstein-Type Polynomials Preserving Shapesp. 114
Shisha-Type Methods and Generalizationsp. 126
Shisha-Type Approximationp. 126
L-Positive Approximationp. 129
Approximation Preserving Three Classical Shapesp. 133
Harmonic Polynomial Approximationp. 133
Subharmonic Polynomial Approximationp. 136
Convex Polynomial Approximationp. 138
Bivariate Monotone Approximation by Convolution Polynomialsp. 154
Tensor Product Polynomials Preserving Popoviciu's Convexitiesp. 160
Bivariate/Multivariate Monotone and Convex Approximationp. 160
Concepts in Bivariate Coshape Approximationp. 178
Bivariate Copositive Approximationp. 186
Bivariate Comonotone Approximationp. 194
Bivariate Shape-Preserving Interpolationp. 207
Bibliographical Notes and Open Problemsp. 209
Shape-Preserving Approximation by Complex Univariate Polynomialsp. 215
Introductionp. 215
Shisha-Type Methods and Generalizationsp. 224
Shisha-Type Approximationp. 230
Re[L]-Positive Approximationp. 235
Shape-Preserving Approximation by Convolution Polynomialsp. 239
Bell-Shaped Kernels and Complex Convolutionsp. 240
Geometric and Approximation Properties of Various Complex Convolutionsp. 247
Approximation and Geometric Properties of Bernstein Polynomialsp. 263
Bibliographical Notes and Open Problemsp. 280
Shape-Preserving Approximation by Complex Multivariate Polynomialsp. 283
Introductionp. 283
Bernstein-Type Polynomials Preserving Univalencep. 286
Shape-Preserving Approximation by Other Types of Polynomialsp. 290
Bibliographical Notes and Open Problemsp. 302
Appendix: Some Related Topicsp. 305
Shape-Preserving Approximation by General Linear Operators on C[a, b]p. 305
Some Real and Complex Nonpolynomial Operators Preserving Shapep. 309
Shape-Preserving Polynomial Approximation in Ordered Vector Spacesp. 312
Complex Nonpolynomial Convolutions Preserving Shapep. 316
Bibliographical Notes and Open Problemsp. 323
Referencesp. 329
Indexp. 351
Table of Contents provided by Ingram. All Rights Reserved.

ISBN: 9780817647025
ISBN-10: 0817647023
Audience: Tertiary; University or College
Format: Hardcover
Language: English
Number Of Pages: 352
Published: 1st June 2008
Publisher: BIRKHAUSER BOSTON INC
Country of Publication: US
Dimensions (cm): 23.5 x 15.5  x 1.91
Weight (kg): 0.71