Intro
Preface
Contents
1 Introduction
2 Basics from Linear and Nonlinear Functional Analysis
2.1 Linear Operators, Banach and Hilbert Spaces
2.2 Fundamental Principles
2.3 Compact Sets and Compact Operators
2.4 Function Spaces
2.4.1 Lp-Spaces
2.4.2 Spaces of Continuous Functions
2.4.3 Approximation Spaces and Unbounded Linear Operators
2.5 Fredholm Operators
2.6 Stability of Operator Sequences
2.7 Fixed Point Theorems and Newton's Method
3 Weighted Polynomial Approximation and Quadrature Rules on ( -1,1)

3.1 Moduli of Smoothness, K-Functionals, and Best Approximation
3.1.1 Moduli of Smoothness and K-Functionals
3.1.2 Moduli of Smoothness and Best Weighted Approximation
3.1.3 Besov-Type Spaces
3.2 Polynomial Approximation with Doubling Weights on the Interval (-1,1)
3.2.1 Definitions
3.2.2 Polynomial Inequalities with Doubling Weights
3.2.3 Christoffel Functions with Respect to Doubling Weights
3.2.4 Convergence of Fourier Sums in Weighted Lp-Spaces
3.2.5 Lagrange Interpolation in Weighted Lp-Spaces
3.2.6 Hermite Interpolation
3.2.7 Hermite-Fejér Interpolation

3.2.8 Lagrange-Hermite Interpolation
3.3 Polynomial Approximation with Exponential Weights on the Interval ( -1,1)
3.3.1 Polynomial Inequalities
3.3.2 K-Functionals and Moduli of Smoothness
3.3.3 Estimates for the Error of Best Weighted Polynomial Approximation
3.3.4 Fourier Sums in Weighted Lp-Spaces
3.3.5 Lagrange Interpolation in Weighted Lp-Spaces
3.3.6 Gaussian Quadrature Rules
4 Weighted Polynomial Approximation and Quadrature Rules on Unbounded Intervals
4.1 Polynomial Approximation with Generalized Freud Weights on the Real Line
4.1.1 The Case of Freud Weights

4.1.2 The Case of Generalized Freud Weights
4.1.3 Lagrange Interpolation in Weighted Lp-Spaces
4.1.4 Gaussian Quadrature Rules
4.1.5 Fourier Sums in Weighted Lp-Spaces
4.2 Polynomial Approximation with Generalized Laguerre Weights on the Half Line
4.2.1 Polynomial Inequalities
4.2.2 Weighted Spaces of Functions
4.2.3 Estimates for the Error of Best Weighted Approximation
4.2.4 Fourier Sums in Weighted Lp-Spaces
4.2.5 Lagrange Interpolation in Weighted Lp-Spaces
4.3 Polynomial Approximation with Pollaczek-Laguerre Weights on the Half Line

4.3.1 Polynomial Inequalities
4.3.2 Weighted Spaces of Functions
4.3.3 Estimates for the Error of Best Weighted Polynomial Approximation
4.3.4 Gaussian Quadrature Rules
4.3.5 Lagrange Interpolation in L2w
4.3.6 Remarks on Numerical Realizations
Computation of the Mhaskar-Rahmanov-Saff Numbers
Numerical Construction of Quadrature Rules
Numerical Examples
Comparison with the Gaussian Rule Based on Laguerre Zeros
5 Mapping Properties of Some Classes of Integral Operators
5.1 Some Properties of the Jacobi Polynomials
5.2 Cauchy Singular Integral Operators