Interpolation Processes

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 Lp.- 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.