Fundamentals of Approximation Theory
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Summary
The field of approximation theory has become so vast that it intersects with every other branch of analysis and plays an increasingly important role in applications in the applied sciences and engineering. Fundamentals of Approximation Theory presents a systematic, in-depth treatment of some basic topics in approximation theory designed to emphasize the rich connections of the subject with other areas of study.
With an approach that moves smoothly from the very concrete to more and more abstract levels, this text provides an outstanding blend of classical and abstract topics. The first five chapters present the core of information that readers need to begin research in this domain. The final three chapters the authors devote to special topics-splined functions, orthogonal polynomials, and best approximation in normed linear spaces- that illustrate how the core material applies in other contexts and expose readers to the use of complex analytic methods in approximation theory.
Each chapter contains problems of varying difficulty, including some drawn from contemporary research. Perfect for an introductory graduate-level class, Fundamentals of Approximation Theory also contains enough advanced material to serve more specialized courses at the doctoral level and to interest scientists and engineers.
Table of Contents
DENSITY THEOREMS
Approximation of Periodic Functions
The Weierstrass Theorem
The Stone-Weierstrass Theorem
LINEAR CHEBYSHEV APPROXIMATION
Approximation in Normed Linear Spaces
Classical Theory
Linear Chebyshev Approximation of Vector-valued Functions
Chebyshev Polynomials
Strong Uniqueness and Continuity of Metric Projection
Discretization
Discrete Best Approximation
The Algorithms of Remes
DEGREE OF APPROXIMATION
Moduli of Continuity
Direct Theorems
Converse Theorems
Approximation by Algebraic Polynomials
Approximation of Analytic Functions
INTERPOLATION
Algebraic Formulation of Finite Interpolation
Lagrange Form
Extended Haar Subspaces and Hermite Interpolation
Hermite-Fejér Interpolation
Divided Differences and the Newton Form
Hermite-Birkhoff Interpolation
FOURIER SERIES
Preliminiaries
Convergence of Fourier Series
Summability
Convergence of Trigonometric Series
Convergence in Mean
SPLINE FUNCTIONS
Preliminaries
Spaces of Piecewise Polynomials and Polynomial Splines
Variational Properties of Spline Interpolants
Construction of Piecewise Polynomial Interpolant
B-Splines
Smoothing Splines
Optimal Quadrature Rules
Generalized Interpolating and Smoothing Spline
Optimal Interpolation
ORTHOGONAL POLYNOMIALS
Jacobi Polynomials
General Properties of Orthagonal Polynomials
Asymptotic Properties
Comments on the Szegö Theory
BEST APPROXIMATION IN NORMED LINEAR SPACES
Approximative Properties of Sets
Characterization and Duality
Continuity of Metric Projections
Convexity, Solarity and Chebyshevity of sets
Best Simultaneous Approximation
Optimal Recovery
BIBLIOGRAPHY
INDEX
o Each chapter also contains an Introduction, Notes and Exercises
Editorial Reviews
"Can this book be used as a text for a course in approximation theory? The answer is a qualified yes. . . In short, a graduate student who was fluent in this book would be prepared for research in almost any areas of approximation theory, with the possible exception of wavelets. Hence the book should be a valuable resource."
-F. Deutsch
"This book presents a systematic, in-depth treatment of some basic topics in approximation theory designed to emphasize the rich connections of the subject with other areas of study. With an approach that moves smoothly from the very concrete to more and more abstract levels, this text provides a blend of classical and abstract topics.... Ideal for an introductory graduate-level class, this book also contains enough advanced material to serve more specialized courses at the doctoral level and to interest scientists and engineers."
-Appl Mech Rev, vol. 54, no. 6, November 2001
