Shape-Preserving Approximation by Real and Complex Polynomials

by Gal, Sorin G.; Anastassiou, George A.

• ISBN13: 9780817647025
• ISBN10: 0817647023
• Edition: 1st
• Format: Hardcover
• Copyright: 2008-10-01
• Publisher: Birkhauser

Summary

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.

Table of Contents

Preface	p. vii
Shape-Preserving Approximation by Real Univariate Polynomials	p. 1
Introduction	p. 1
Shape-Preserving Interpolation by Polynomials	p. 7
Bernstein-Type Polynomials Preserving Shapes	p. 19
Shisha-Type Results	p. 35
Positive and Copositive Polynomial Approximation	p. 38
Pointwise Positive Approximation	p. 38
L[superscript p]-Positive Approximation, 0 < p < [infinity]	p. 39
Uniform and Pointwise Copositive Approximation	p. 41
L[superscript p]-Copositive Approximation, 0 < p < [infinity]	p. 47
Copositive Approximation with Modified Weighted Moduli of Smoothness	p. 48
Generalizations	p. 50
Monotone and Comonotone Polynomial Approximation	p. 54
L[superscript p]-Monotone Approximation, 0 < p [less than or equal infinity]	p. 57
Pointwise Monotone Approximation	p. 62
L[superscript p]-Comonotone Approximation, 0 < p [less than or equal infinity]	p. 64
Comonotone Approximation with Modified Weighted Moduli of Smoothness	p. 68
Nearly Comonotone Approximation	p. 70
Convex and Coconvex Polynomial Approximation	p. 73
Linear Methods in Convex Approximation	p. 74
Nonlinear Methods in Convex Approximation	p. 80
Pointwise Convex Approximation	p. 81
Convex Approximation with Modified Weighted Moduli of Smoothness	p. 82
Uniform Coconvex Approximation	p. 83
Coconvex Approximation with Modified Weighted Moduli of Smoothness	p. 85
Pointwise Coconvex Approximation	p. 86
Nearly Coconvex Approximation	p. 87
Shape-Preserving Approximation by Convolution Polynomials	p. 90
Positive Linear Polynomial Operators Preserving Shape	p. 94
Notes	p. 95
Shape-Preserving Approximation by Real Multivariate Polynomials	p. 99
Introduction	p. 99
Bernstein-Type Polynomials Preserving Shapes	p. 114
Shisha-Type Methods and Generalizations	p. 126
Shisha-Type Approximation	p. 126
L-Positive Approximation	p. 129
Approximation Preserving Three Classical Shapes	p. 133
Harmonic Polynomial Approximation	p. 133
Subharmonic Polynomial Approximation	p. 136
Convex Polynomial Approximation	p. 138
Bivariate Monotone Approximation by Convolution Polynomials	p. 154
Tensor Product Polynomials Preserving Popoviciu's Convexities	p. 160
Bivariate/Multivariate Monotone and Convex Approximation	p. 160
Concepts in Bivariate Coshape Approximation	p. 178
Bivariate Copositive Approximation	p. 186
Bivariate Comonotone Approximation	p. 194
Bivariate Shape-Preserving Interpolation	p. 207
Bibliographical Notes and Open Problems	p. 209
Shape-Preserving Approximation by Complex Univariate Polynomials	p. 215
Introduction	p. 215
Shisha-Type Methods and Generalizations	p. 224
Shisha-Type Approximation	p. 230
Re[L]-Positive Approximation	p. 235
Shape-Preserving Approximation by Convolution Polynomials	p. 239
Bell-Shaped Kernels and Complex Convolutions	p. 240
Geometric and Approximation Properties of Various Complex Convolutions	p. 247
Approximation and Geometric Properties of Bernstein Polynomials	p. 263
Bibliographical Notes and Open Problems	p. 280
Shape-Preserving Approximation by Complex Multivariate Polynomials	p. 283
Introduction	p. 283
Bernstein-Type Polynomials Preserving Univalence	p. 286
Shape-Preserving Approximation by Other Types of Polynomials	p. 290
Bibliographical Notes and Open Problems	p. 302
Appendix: Some Related Topics	p. 305
Shape-Preserving Approximation by General Linear Operators on C[a, b]	p. 305
Some Real and Complex Nonpolynomial Operators Preserving Shape	p. 309
Shape-Preserving Polynomial Approximation in Ordered Vector Spaces	p. 312
Complex Nonpolynomial Convolutions Preserving Shape	p. 316
Bibliographical Notes and Open Problems	p. 323
References	p. 329
Index	p. 351
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