Interpolation Processes [electronic resource] : Basic Theory and Applications /
Mastroianni, Giuseppe.author.
Milovanovic, Gradimir.author.
The classical books on interpolation address numerous negative results, i.e., results on divergent interpolation processes, usually constructed over some equidistant systems of nodes. The authors present, with complete proofs, recent results on convergent interpolation processes, for trigonometric and algebraic polynomials of one real variable, not yet published in other textbooks and monographs on approximation theory and numerical mathematics. In this special, but fundamental and important field of real analysis the authors present the state of art. Some 500 references are cited, including many new results of the authors. Basic tools in this field (orthogonal polynomials, moduli of smoothness, K-functionals, etc.) as well as some selected applications in numerical integration, integral equations, moment-preserving approximation and summation of slowly convergent series are also given. Beside the basic properties of the classical orthogonal polynomials the book provides new results on nonclassical orthogonal polynomials including methods for their numerical construction.
Description based upon print version of record.
Includes bibliographical references and index.
1. Constructive Elements and Approaches in Approximation Theory -- 1.1 Introduction to Approximation Theory -- 1.2 Basic Facts on Trigonometric Approximation -- 1.3 Chebyshev Systems and Interpolation -- 1.4 Interpolation by Algebraic Polynomials -- 2. Orthogonal Polynomials and Weighted Polynomial Approximation -- 2.1 Orthogonal Systems and Polynomials -- 2.2 Orthogonal Polynomials on the Real Line -- 2.3 Classical Orthogonal Polynomials -- 2.4 Nonclassical Orthogonal Polynomials -- 2.5 Weighted Polynomial Approximation -- 3. Trigonometric Approximation -- 3.1 Approximating Properties of Operators -- 3.2 Discrete Operators -- 4. Algebraic Interpolation in Uniform Norm -- 4.1 Introduction and Preliminaries -- 4.2 Optimal Systems of Nodes -- 4.3 Weighted Interpolation -- 5. Applications -- 5.1 Quadrature Formulae -- 5.2 Integral Equations -- 5.3 Moment-Preserving Approximation -- 5.4 Summation of Slowly Convergent Series -- References -- Index.
The classical books on interpolation address numerous negative results, i.e., results on divergent interpolation processes, usually constructed over some equidistant systems of nodes. The authors present, with complete proofs, recent results on convergent interpolation processes, for trigonometric and algebraic polynomials of one real variable, not yet published in other textbooks and monographs on approximation theory and numerical mathematics. In this special, but fundamental and important field of real analysis the authors present the state of art. Some 500 references are cited, including many new results of the authors. Basic tools in this field (orthogonal polynomials, moduli of smoothness, K-functionals, etc.) as well as some selected applications in numerical integration, integral equations, moment-preserving approximation and summation of slowly convergent series are also given. Beside the basic properties of the classical orthogonal polynomials the book provides new results on nonclassical orthogonal polynomials including methods for their numerical construction.
Integral equations.
Special functions.
Sequences (Mathematics).
Fourier analysis.
Integral Equations.
Special Functions.
Sequences, Series, Summability.
Fourier Analysis.
Springer Monographs in Mathematics