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Interpolation Processes: Basic Theory and Applications (Springer Monographs in Mathematics)by Giuseppe Mastroianni
Synopses & ReviewsPublisher Comments: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, Kfunctionals, etc.) as well as some selected applications in numerical integration, integral equations, momentpreserving 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.
Synopsis:This book presents, with complete proofs, recent results on convergent interpolation processes, for trigonometric and algebraic polynomials of one real variable, not yet published in other books and monographs on approximation theory and numerical mathematics.
About the AuthorGradimir V. Milovanovic is Professor of the University of Niš and Corresponding member of the Serbian Academy of Sciences and Arts.
Table of Contents1. Constructive Elements and Approaches in Approximation Theory. 1.1 Introduction to Approximation Theory. 1.1.1 Basic notions. 1.1.2 Algebraic and trigonometric polynomials. 1.1.3 Best approximation by polynomials. 1.1.4 Chebyshev polynomials. 1.1.5 Chebyshev extremal problems. 1.1.6 Chebyshev alternation theorem. 1.1.7 Numerical methods. 1.2 Basic Facts on Trigonometric Approximation. 1.2.1 Trigonometric kernels. 1.2.2 Fourier series and sums. 1.2.3 Moduli of smoothness, best approximation and Besov spaces. 1.3 Chebyshev Systems and Interpolation. 1.3.1 Chebyshev systems and spaces. 1.3.2 Algebraic Lagrange interpolation. 1.3.3 Trigonometric interpolation. 1.3.4 Riesz interpolation formula. 1.3.5 A general interpolation problem. 1.4 Interpolation by Algebraic Polynomials. 1.4.1 Representations and computation of interpolation polynomials. 1.4.2 Interpolation array and Lagrange operators. 1.4.3 Interpolation error for some classes of functions. 1.4.4 Uniform convergence in the class of analytic functions. 1.4.5 Bernstein's example of pointwise divergence. 1.4.6 Lebesgue function and some estimates for the Lebesgue constant. 1.4.7 Algorithm for finding optimal nodes. 2. Orthogonal Polynomials and Weighted Polynomial Approximation. 2.1 Orthogonal Systems and Polynomials. 2.1.1 Inner product space and orthogonal systems. 2.1.2 Fourier expansion and best approximation. 2.1.3 Examples of orthogonal systems. 2.1.4 Basic facts on orthogonal polynomials and extremal problems. 2.1.5 Zeros of orthogonal polynomials. 2.2 Orthogonal Polynomials on the Real Line. 2.2.1 Basic properties. 2.2.2 Asymptotic properties of orthogonal polynomials. 2.2.3 Associated polynomials and Christoffel numbers. 2.2.4 Functions of the second kind and Stieltjes polynomials. 2.3 Classical Orthogonal Polynomials. 2.3.1 Definition of the classical orthogonal polynomials. 2.3.2 General properties of the classical orthogonal polynomials. 2.3.3 Generating function. 2.3.4 Jacobi polynomials. 2.3.5 Generalized Laguerre polynomials. 2.3.6 Hermite polynomials. 2.4 Nonclassical Orthogonal Polynomials. 2.4.1 Semiclassical orthogonal polynomials. 2.4.2 Generalized Gegenbauer polynomials. 2.4.3 Generalized Jacobi polynomials. 2.4.4 SoninMarkov orthogonal polynomials. 2.4.5 Freud orthogonal polynomials. 2.4.6 Orthogonal polynomials with respect to Abel, Lindelöf, and logistic weights. 2.4.7 Strong nonclassical orthogonal polynomials. 2.4.8 Numerical construction of orthogonal polynomials. 2.5 Weighted Polynomial Approximation. 2.5.1 Weighted functional spaces, moduli of smoothness and Kfunctionals. 2.5.2 Weighted best polynomial approximation on (1,1). 2.5.3 Weighted approximation on the semiaxis. 2.5.4 Weighted approximation on the real line. 2.5.5 Weighted polynomial approximation of functions having isolated interior singularities. 3. Trigonometric Approximation. 3.1 Approximating Properties of Operators. 3.1.1 Approximation by Fourier sums. 3.1.2 Approximation by Fejér and de la Vallée Poussin means. 3.2 Discrete Operators. 3.2.1 A quadrature formula. 3.2.2 Discrete versions of Fourier and de la Vallée Poussin sums. 3.2.3 Marcinkiewicz inequalities. 3.2.4 Uniform approximation. 3.2.5 Lagrange interpolation error in Lp. 3.2.6 Some estimates of the interpolation errors in L1Sobolev spaces. 3.2.7 The weighted case. 4. Algebraic Interpolation in Uniform Norm. 4.1 Introduction and Preliminaries. 4.1.1 Interpolation at zeros of orthogonal polynomials. 4.1.2 Some auxiliary results. 4.2 Optimal Systems of Nodes. 4.2.1 Optimal systems of knots on (1,1). 4.2.2 Additional nodes method with Jacobi zeros. 4.2.3 Other "optimal" interpolation processes. 4.2.4 Some simultaneous interpolation processes. 4.3 Weighted Interpolation. 4.3.1 Weighted interpolation at Jacobi zeros. 4.3.2 Lagrange interpolation in Sobolev spaces. 4.3.3 Interpolation at Laguerre zeros. 4.3.4 Interpolation at Hermite zeros. 4.3.5 Interpolation of functions with internal isolated singularities. 5. Applications. 5.1 Quadrature Formulae. 5.1.1 Introduction. 5.1.2 Some remarks on NewtonCotes rules with Jacobi weights. 5.1.3 GaussChristoffel quadrature rules. 5.1.4 GaussRadau and GaussLobatto quadrature rules. 5.1.5 Error estimates of Gaussian rules for some classes of functions. 5.1.6 Product integration rules. 5.1.7 Integration of periodic functions on the real line with rational weight. 5.2 Integral Equations. 5.2.1 Some basic facts. 5.2.2 Fredholm integral equations of the second kind. 5.2.3 Nyström method. 5.3 MomentPreserving Approximation. 5.3.1 The standard L2approximation. 5.3.2 The constrained L2polynomial approximation. 5.3.3 Momentpreserving spline approximation. 5.4 Summation of Slowly Convergent Series. 5.4.1 Laplace transform method. 5.4.2 Contour integration over a rectangle. 5.4.3 Remarks on some slowly convergent power series. References. Index.
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