Interpolation Processes: Basic Theory and Applications (Springer Monographs in Mathematics)

Preface
Contents
Constructive Elements and Approaches in Approximation Theory
	Introduction to Approximation Theory
		Basic Notions
		Algebraic and Trigonometric Polynomials
		Best Approximation by Polynomials
		Chebyshev Polynomials
			Basic Properties
			Differential Equation
			Zeros and Extremal Points
			Chebyshev Polynomials in the Complex Plane
			Some Other Relations
			Orthogonality
		Chebyshev Extremal Problems
			The Extremal Problem in the Uniform Norm
			The Extremal Problem in L1-norm
		Chebyshev Alternation Theorem
			Some Classical Special Cases
		Numerical Methods
	Basic Facts on Trigonometric Approximation
		Trigonometric Kernels
		Fourier Series and Sums
		Moduli of Smoothness, Best Approximation and Besov Spaces
	Chebyshev Systems and Interpolation
		Chebyshev Systems and Spaces
		Algebraic Lagrange Interpolation
		Trigonometric Interpolation
		Riesz Interpolation Formula
		A General Interpolation Problem
	Interpolation by Algebraic Polynomials
		Representations and Computation of Interpolation Polynomials
		Interpolation Array and Lagrange Operators
		Interpolation Error for Some Classes of Functions
			The Error in the Class of Continuous-Differentiable Functions
			The Error in the Class of Analytic Functions
		Uniform Convergence in the Class of Analytic Functions
		Bernstein\'s Example of Pointwise Divergence
		Lebesgue Function and Some Estimates for the Lebesgue Constant
			Equidistant Nodes
			Chebyshev Nodes
		Algorithm for Finding Optimal Nodes
Orthogonal Polynomials and Weighted Polynomial Approximation
	Orthogonal Systems and Polynomials
		Inner Product Space and Orthogonal Systems
		Fourier Expansion and Best Approximation
		Examples of Orthogonal Systems
			Trigonometric System
			Chebyshev Polynomials
			Orthogonal Polynomials on the Unit Circle
			Orthogonal Polynomials on the Unit Disk
			Orthogonal Polynomials on the Ellipse
			Malmquist-Takenaka System of Rational Functions
			Polynomials Orthogonal on the Radial Rays
			Müntz Orthogonal Polynomials
			Müntz Orthogonal Polynomials of the Second Kind
			Generalized Exponential Polynomials
			Discrete Chebyshev Polynomials
			Formal Orthogonal Polynomials with Respect to a Moment Functional
		Basic Facts on Orthogonal Polynomials and Extremal Problems
		Zeros of Orthogonal Polynomials
	Orthogonal Polynomials on the Real Line
		Basic Properties
			Three-Term Recurrence Relations
			Christoffel\'s Formulae
			Zeros
			Some Special Weights
		Asymptotic Properties of Orthogonal Polynomials
			Bernstein-Szego Identities
			The Fokas-Its-Kitaev (Riemann-Hilbert) Identity
			Rakhmanov\'s Identity
		Associated Polynomials and Christoffel Numbers
			Associated Polynomials
			Stieltjes Transform of the Measure and Christoffel Numbers
			Markov\'s Moment Problem
		Functions of the Second Kind and Stieltjes Polynomials
	Classical Orthogonal Polynomials
		Definition of the Classical Orthogonal Polynomials
		General Properties of the Classical Orthogonal Polynomials
		Generating Function
		Jacobi Polynomials
			Special Cases
			Zeros
			Inequalities and Asymptotics
			Christoffel Function and Christoffel Numbers
		Generalized Laguerre Polynomials
			Zeros
			Inequalities
			Christoffel Function and Christoffel Numbers
		Hermite Polynomials
	Nonclassical Orthogonal Polynomials
		Semi-classical Orthogonal Polynomials
		Generalized Gegenbauer Polynomials
		Generalized Jacobi Polynomials
		Sonin-Markov Orthogonal Polynomials
		Freud Orthogonal Polynomials
			Mhaskar-Rakhmanov-Saff Number
			Basic Properties of Freud Polynomials
			Strong Asymptotics
		Orthogonal Polynomials with Respect to Abel, Lindelöf, and Logistic Weights
		Strong Non-classical Orthogonal Polynomials
		Numerical Construction of Orthogonal Polynomials
			Modified Chebyshev Algorithm
			Discretized Stieltjes-Gautschi Procedure
	Weighted Polynomial Approximation
		Weighted Functional Spaces, Moduli of Smoothness and K-functionals
		Weighted Best Polynomial Approximation on [-1,1]
		Weighted Approximation on the Semi-axis
			Weighted K-functionals and Moduli of Smoothness
			Weighted Best Polynomial Approximation
			Weighted Besov Type Spaces
		Weighted Approximation on the Real Line
		Weighted Polynomial Approximation of Functions Having Isolated Interior Singularities
Trigonometric Approximation
	Approximating Properties of Operators
		Approximation by Fourier Sums
		Approximation by Fejér and de la Vallée Poussin Means
	Discrete Operators
		A Quadrature Formula
		Discrete Versions of Fourier and de la Vallée Poussin Sums
		Marcinkiewicz Inequalities
		Uniform Approximation
		Lagrange Interpolation Error in Lp
		Some Estimates of the Interpolation Errors in L1-Sobolev Spaces
		The Weighted Case
Algebraic Interpolation in Uniform Norm
	Introduction and Preliminaries
		Interpolation at Zeros of Orthogonal Polynomials
		Some Auxiliary Results
	Optimal Systems of Nodes
		Optimal Systems of Knots on [-1,1]
			Interpolation at Jacobi Abscissas
			Interpolation at the ``Practical Abscissas\'\'
		Additional Nodes Method with Jacobi Zeros
		Other ``Optimal\'\' Interpolation Processes
			Interpolation with Associated Polynomials
			Interpolation at Stieltjes Zeros
			Extended Interpolation
		Some Simultaneous Interpolation Processes
	Weighted Interpolation
		Weighted Interpolation at Jacobi Zeros
		Lagrange Interpolation in Sobolev Spaces
		Interpolation at Laguerre Zeros
		Interpolation at Hermite Zeros
		Interpolation of Functions with Internal Isolated Singularities
			Interpolation Processes on Bounded Intervals
			Interpolation Processes on Unbounded Intervals
			Numerical Examples
Applications
	Quadrature Formulae
		Introduction
		Some Remarks on Newton-Cotes Rules with Jacobi Weights
		Gauss-Christoffel Quadrature Rules
			Gauss-Christoffel Quadratures for the Classical Weights
			Computation of Gauss-Christoffel Quadratures
		Gauss-Radau and Gauss-Lobatto Quadrature Rules
			Gauss-Radau Quadrature Formula
			Gauss-Lobatto Quadrature Formula
		Error Estimates of Gaussian Rules for Some Classes of Functions
			Error Estimates for Analytic Functions
			Error Estimates for Some Classes of Continuous Functions
			Error Estimates for Gauss-Laguerre Formula
			Error Estimates for Freud-Gaussian Rules
		Product Integration Rules
		Integration of Periodic Functions on the Real Line with Rational Weight
	Integral Equations
		Some Basic Facts
		Fredholm Integral Equations of the Second Kind
			Locally Smooth Kernels
			Numerical Examples
			Weakly Singular Kernels
		Nyström Method
	Moment-Preserving Approximation
		The Standard L2-Approximation
			Generalization
		The Constrained L2-Polynomial Approximation
		Moment-Preserving Spline Approximation
			Approximation on [0,+)
			Approximation on a Compact Interval
	Summation of Slowly Convergent Series
		Laplace Transform Method
		Contour Integration Over a Rectangle
		Remarks on Some Slowly Convergent Power Series
References
Index