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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 this eld (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.

Interpolation Processes: Basic Theory and Applications

1. 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 Semi-classical orthogonal polynomials.- 2.4.2 Generalized Gegenbauer polynomials.- 2.4.3 Generalized Jacobi polynomials.- 2.4.4 Sonin-Markov 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 non-classical 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 K-functionals.- 2.5.2 Weighted best polynomial approximation on (-1,1).- 2.5.3 Weighted approximation on the semi-axis.- 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 L^p.- 3.2.6 Some estimates of the interpolation errors in L¹-Sobolev 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 Newton-Cotes rules with Jacobi weights.- 5.1.3 Gauss-Christoffel quadrature rules.- 5.1.4 Gauss-Radau and Gauss-Lobatto 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 Moment-Preserving Approximation.- 5.3.1 The standard L²-approximation.- 5.3.2 The constrained L²-polynomial approximation.- 5.3.3 Moment-preserving 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.