Interpolation Processes

Interpolation of functions is one of the basic part of Approximation Theory. There are many books on approximation theory, including interpolation methods that - peared in the last fty years, but a few of them are devoted only to interpolation processes. An example is the book of J. Szabados and P. Vértesi: Interpolation of Functions, published in 1990 by World Scienti c. Also, two books deal with a special interpolation problem, the so-called Birkhoff interpolation, written by G.G. Lorentz, K. Jetter, S.D. Riemenschneider (1983) and Y.G. Shi (2003). The classical books on interpolation address numerous negative results, i.e., - sultsondivergentinterpolationprocesses,usuallyconstructedoversomeequidistant system of nodes. The present book deals mainly with new results on convergent - terpolation processes in uniform norm, for algebraic and trigonometric polynomials, not yet published in other textbooks and monographs on approximation theory and numerical mathematics. Basic tools in thiseld (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. The rstchapterprovidesanaccountofbasicfactsonapproximationbyalgebraic and trigonometric polynomials introducing the most important concepts on appro- mation of functions. Especially, in Sect. 1.4 we give basic results on interpolation by algebraic polynomials, including representations and computation of interpolation polynomials, Lagrange operators, interpolation errors and uniform convergence in some important classes of functions, as well as an account on the Lebesgue function and some estimates for the Lebesgue constant.

There are many books on approximation theory, including interpolation methods that appeared in the last fifty years, but a few of them are devoted only to interpolation processes as is this book Includes supplementary material: sn.pub/extras

Autorentext

Peter Junghanns is Professor Emeritus at the Mathematics Department of the Technical University of Chemnitz, Germany, where he got his habilitation in 1984. His field of specialization includes analysis and numerical methods for operator equations, in particular integral equations. Giuseppe Mastroianni is emeritus professor at the University of Basilicata (Italy) since 2013, where he had been full professor of Numerical Analysis since 1987. In 1971 he graduated in Mathematics at the University of Naples "Federico II", where he began his academic career. He is author of numerous papers on polynomial approximation, positive operators, mechanical quadrature and numerical treatment of integral equations as well as of the monographs "Interpolation processes. Basic theory and applications" (with G.V. Milovanovic) and "Elementi di teoria dell'approssimazione polinomiale" (with M.C. De Bonis and I. Notarangelo). Incoronata Notarangelo is assistant professor of Numerical Analysis at the University of Turin (Italy) since 2019. In 2010 she received a PhD in Mathematics from the University of Basilicata (Italy), in cooperation with the University of Szeged (Hungary). She is author of papers on polynomial approximation, quadrature rules and numerical methods for integral equations as well as of the monograph "Elementi di teoria dell'approssimazione polinomiale" (with M.C. De Bonis and G. Mastroianni).

Klappentext

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 resultson nonclassical orthogonal polynomials including methods for their numerical construction.

Inhalt

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.