Weighted Polynomial Approximation and Numerical Methods for Integral Equations

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
		5.2.1 Weighted L2-Spaces
		5.2.2 Weighted Spaces of Continuous Functions
		5.2.3 On the Case of Variable Coefficients
		5.2.4 Regularity Properties
	5.3 Compact Integral Operators
	5.4 Weakly Singular Integral Operators with Logarithmic Kernels
	5.5 Singular Integro-Differential or Hypersingular Operators
	5.6 Operators with Fixed Singularities of Mellin Type
	5.7 A Note on the Invertibility of Singular Integral Operators with Cauchy and Mellin Kernels
	5.8 Solvability of Nonlinear Cauchy Singular Integral Equations
		5.8.1 Equations of the First Type
		5.8.2 Equations of the Second Type
		5.8.3 Equations of the Third Type
6 Numerical Methods for Fredholm Integral Equations
	6.1 Collectively Compact Sequences of Integral Operators
	6.2 The Classical Nyström Method
		6.2.1 The Case of Jacobi Weights
		6.2.2 The Case of an Exponential Weight on (0,∞)
		6.2.3 The Application of Truncated Quadrature Rules
	6.3 The Nyström Method Based on Product Integration Formulas
		6.3.1 The Case of Jacobi Weights
		6.3.2 The Case of an Exponential Weight on (0,∞)
		6.3.3 Application to Weakly Singular Integral Equations
	6.4 Integral Equations with Logarithmic Kernels
		6.4.1 The Well-posed Case
		6.4.2 The Ill-posed Case
		6.4.3 A Collocation-Quadrature Method
		6.4.4 A Fast Algorithm
7 Collocation and Collocation-Quadrature Methods for Strongly Singular Integral Equations
	7.1 Cauchy Singular Integral Equations on an Interval
		7.1.1 Collocation and Collocation-Quadrature Methods
		7.1.2 Weighted Uniform Convergence
			Collocation Methods
			Collocation-Quadrature Methods
		7.1.3 Fast Algorithms
			Weighted L2-Convergence
			Computational Complexity of the Algorithm
			Weighted Uniform Convergence
	7.2 Hypersingular Integral Equations
		7.2.1 Collocation and Collocation-Quadrature Methods
		7.2.2 A Fast Algorithm
			First Step of the Algorithm
			Second Step of the Algorithm
			A More General Situation
	7.3 Integral Equations with Mellin Type Kernels
	7.4 Nonlinear Cauchy Singular Integral Equations
		7.4.1 Asymptotic of the Solution
		7.4.2 A Collocation-Quadrature Method
		7.4.3 Convergence Analysis
		7.4.4 A Further Class of Nonlinear Cauchy Singular Integral Equations
			A Collocation-Quadrature Method
			Convergence Analysis
8 Applications
	8.1 A Cruciform Crack Problem
		8.1.1 The Integral Equations Under Consideration
		8.1.2 Solvability Properties of the Operator Equations
			Equation (I+MH0)u0=f0
			Equation (I+MH1)u1=f1
			Equation (I+H2)u2=f2
		8.1.3 A Quadrature Method
	8.2 The Drag Minimization Problem for a Wing
		8.2.1 Formulation of the Problem
		8.2.2 Derivation of the Operator Equation
		8.2.3 A Collocation-Quadrature Method
		8.2.4 Numerical Examples
	8.3 Two-Dimensional Free Boundary Value Problems
		8.3.1 Seepage Flow from a Dam
			The Linear Case
			The Nonlinear Case
		8.3.2 Seepage Flow from a Channel
			Generating Gaussian Rules
			The Application of Product Integration Rules
			Numerical Results
9 Hints and Answers to the Exercises
10 Equalities and Inequalities
	10.1 Equalities and Equivalences
	10.2 General Inequalities
	10.3 Marcinkiewicz Inequalities
Bibliography
Index