Overview

This volume, and its predecessor in the series, cover the whole field of approximation of functions of one real variable. The main subject of this volume is approximation by polymonials, rational functions, splines and operators. There are excursions into the related fields of interpolation, complex variable approximation, wavelets, widths and functional analysis. Emphasis is placed on basic results and illustrative examples, rather than on generality or special problems.

Full Product Details

Author:   G. G. Lorentz ,  Manfred V. Golitschek ,  Yuly Makovoz ,  Y. Makovoz (Lowell University, USA)
Publisher:   Springer-Verlag Berlin and Heidelberg GmbH & Co. KG
Imprint:   Springer-Verlag Berlin and Heidelberg GmbH & Co. K
Volume:   v. 304
Weight:   1.105kg
ISBN:  

9783540570288

ISBN 10:   3540570284
Pages:   660
Publication Date:   14 May 1996
Audience:   College/higher education ,  Professional and scholarly ,  Postgraduate, Research & Scholarly ,  Professional & Vocational
Format:   Hardback
Publisher's Status:   Active
Availability:   Out of stock  
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Table of Contents

1. Problems of Polynomial Approximation.- 1. Examples of Polynomials of Best Approximation.- 2. Distribution of Alternation Points of Polynomials of Best Approximation.- 3. Distribution of Zeros of Polynomials of Best Approximation.- 4. Error of Approximation.- 5. Approximation on (-?, ?) by Linear Combinations of Functions (x - c)-1.- 6. Weighted Approximation by Polynomials on (-?, ?).- 7. Spaces of Approximation Theory.- 8. Problems and Notes.- 2. Approximation Problems with Constraints.- 1. Introduction.- 2. Growth Restrictions for the Coefficients.- 3. Monotone Approximation.- 4. Polynomials with Integral Coefficients.- 5. Determination of the Characteristic Sets.- 6. Markov-Type Inequalities.- 7. The Inequality of Remez.- 8. One-sided Approximation by Polynomials.- 9. Problems.- 10. Notes.- 3. Incomplete Polynomials.- 1. Incomplete Polynomials.- 2. Incomplete Chebyshev Polynomials.- 3. Incomplete Trigonometric Polynomials.- 4. Sequences of Polynomials with Many Real Zeros.- 5. Problems.- 6. Notes.- 4. Weighted Polynomials.- 1. Essential Sets of Weighted Polynomials.- 2. Weighted Chebyshev Polynomials.- 3. The Equilibrium Measure.- 4. Determination of Minimal Essential Sets.- 5. Weierstrass Theorems and Oscillations.- 6. Weierstrass Theorem for Freud Weights.- 7. Problems.- 8. Notes.- 5. Wavelets and Orthogonal Expansions.- 1. Multiresolutions and Wavelets.- 2. Scaling Functions with a Monotone Majorant.- 3. Periodization.- 4. Polynomial Schauder Bases.- 5. Orthonormal Polynomial Bases.- 6. Problems and Notes.- 6. Splines.- 1. General Facts.- 2. Splines of Best Approximation.- 3. Periodic Splines.- 4. Convergence of Some Spline Operators.- 5. Notes.- 7. Rational Approximation.- 1. Introduction.- 2. Best Rational Approximation.- 3. Rational Approximation of |x|.- 4. Approximation of e-xon [-1,1].- 5. Rational Approximation of e-x on [0, ?).- 6. Approximation of Classes of Functions.- 7. Theorems of Popov.- 8. Properties of the Operator of Best Rational Approximation in C and Lp.- 9. Approximation by Rational Functions with Arbitrary Powers.- 10. Problems.- 11. Notes.- 8. StahPs Theorem.- 1. Introduction and Main Result.- 2. A Dirichlet Problem on [1/2, l/pn].- 3. The Second Approach to the Dirichlet Problem.- 4. Proof of Theorem 1.1.- 5. Notes.- 9. Pade Approximation.- 1. The Pade Table.- 2. Convergence of the Rows of the Pade Table.- 3. The Nuttall-Pommerenke Theorem.- 4. Problems.- 5. Notes.- 10. Hardy Space Methods in Rational Approximation.- 1. Bernstein-Type Inequalities for Rational Functions.- 2. Uniform Rational Approximation in Hardy Spaces.- 3. Approximation by Simple Functions.- 4. The Jackson-Rusak Operator; Rational Approximation of Sums of Simple Functions.- 5. Rational Approximation on T and on [-1,1].- 6. Relations Between Spline and Rational Approximation in the Spaces 0 ?.- 7. Problems.- 8. Notes.- 11. Muntz Polynomials.- 1. Definitions and Simple Properties.- 2. Muntz-Jackson Theorems.- 3. An Inverse Muntz-Jackson Theorem.- 4. The Index of Approximation.- 5. Markov-Type Inequality for Muntz Polynomials.- 6. Problems.- 7. Notes.- 12. Nonlinear Approximation.- 1. Definitions and Simple Properties.- 2. Varisolvent Families.- 3. Exponential Sums.- 4. Lower Bounds for Errors of Nonlinear Approximation.- 5. Continuous Selections from Metric Projections.- 6. Approximation in Banach Spaces: Suns and Chebyshev Sets.- 7. Problems.- 8. Notes.- 13. Widths I.- 1. Definitions and Basic Properties.- 2. Relations Between Different Widths.- 3. Widths of Cubes and Octahedra.- 4. Widths in Hilbert Spaces.- 5. Applications of Borsuk's Theorem.- 6. Variational Problems and Spectral Functions.- 7. Results of Buslaev and Tikhomirov.- 8. Classes of Differentiate Functions on an Interval.- 9. Classes of Analytic Functions.- 10. Problems.- 11. Notes.- 14. Widths II: Weak Asymptotics for Widths of Lipschitz Balls, Random Approximants.- 1. Introduction.- 2. Discretization.- 3. Weak Equivalences for Widths. Elementary Methods.- 4. Distribution of Scalar Products of Unit Vectors.- 5. Kashin's Theorems.- 6. Gaussian Measures.- 7. Linear Widths of Finite Dimensional Balls.- 8. Linear Widths of the Lipschitz Classes.- 9. Problems.- 10. Notes.- 15. Entropy.- 1. Entropy and Capacity.- 2. Elementary Estimates.- 3. Linear Approximation and Entropy.- 4. Relations Between Entropy and Widths.- 5. Entropy of Classes of Analytic Functions.- 6. The Birman-Solomyak Theorem.- 7. Entropy Numbers of Operators.- 8. Notes.- 16. Convergence of Sequences of Operators.- 1. Introduction.- 2. Simple Necessary and Sufficient Conditions.- 3. Geometric Properties of Dominating Sets.- 4. Strict Dominating Systems; Minimal Systems; Examples.- 5. Shadows of Sets of Continuous Functions.- 6. Shadows in Banach Function Spaces.- 7. Positive Contractions.- 8. Contractions.- 9. Notes.- 17. Representation of Functions by Superpositions.- 1. The Theorems of Kolmogorov.- 2. Proof of the Theorems.- 3. Functions Not Representable by Superpositions.- 4. Linear Superpositions.- 5. Notes.- Appendix 1. Theorems of Borsuk and of Brunn-Minkowski.- 1. Borsuk's Theorem.- 2. The Brunn-Minkowski Inequality.- Appendix 2. Estimates of Some Elliptic Integrals.- Appendix 3. Hardy Spaces and Blaschke Products.- 1. Hardy Spaces.- 2. Conjugate Functions and Cauchy Integrals.- 3. Atomic Decompositions in Hardy Spaces.- 4. Blaschke Products.- Appendix 4. Potential Theory and Logarithmic Capacity.- 1. Logarithmic Potentials.- 2. Equilibrium Distribution and Logarithmic Capacity.- 3. The Dirichlet Problem and Green's Function.- 4. Balayage Methods.- Author Index.

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