Preface

Interpolation of functions is one of the basic part of Approximation Theory. 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. An example is the book of J. Szabados and P. Vértesi: Interpolation of Functions, published in 1990 by World Scientific. 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., results on divergent interpolation processes, usually constructed over some equidistant system of nodes. The present book deals mainly with new results on convergent interpolation 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 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. The first chapter provides an account of basic facts on approximation by algebraic and trigonometric polynomials introducing the most important concepts on approximation 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. The second chapter is devoted to orthogonal polynomials on the real line and weighted polynomial approximation. For polynomials orthogonal on the real line we give the basic properties and introduce and discuss the associated polynomials, functions of the second kind, Stieltjes polynomials, as well as the Christoffel functions and numbers. The classical orthogonal polynomials as the most important class of orthogonal polynomials on the real line are treated in Sect. 2.3, and new results on nonclassical orthogonal polynomials, including methods for their numerical construction, are studied in Sect. 2.4. Introducing the weighted functional spaces, moduli of smoothness and K-functionals, the weighted best polynomial approximations on (−1,1), (0,+∞) and (−∞,+∞) are treated in Sect. 2.5, as well as the weighted polynomial approximation of functions having interior isolated singularities. Trigonometric approximation is considered in Chap. 3. Approximations by sums of Fourier and Fejér and de la Vallée Poussin means are given. Their discrete versions and the Lagrange trigonometric operator are also investigated. As a basic tool for studying approximating properties of the Lagrange and de la Vallée Poussin operators we consider the so-called Marcinkiewicz inequalities. Beside the uniform