CiNii Books - Fundamentals of approximation theory

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-Fejer 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 Szegoe 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

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Fundamentals of approximation theory

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