Constructive Function Theory, Volume II: Approximation in Mean - ISBN:0000000000

Constructive Function Theory, Volume II: Approximation in Mean

Date: 23 April 2011, 23:49

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Chapter I. THE SPACE L(z)
1. The Problem
2. The Weight Function. The Space L;(z)
3. Convergence in Mean
4. Classes of Functions Dense in L(z)
Chapter II. ORTHOGONAL SYSTEMS
1. Orthogonality. Examples
2. The FOURIER Coefficients
3. Completeness and Closedness
Chapter III. LINEARLY INDEPENDENT SYSTEMS OF FUNCTIONS
1. Linear Independence. The GRAM Determinant. The SCHMIDT Theorem
2. Approximation by Linearly Independent Functions
3. The MUNTZ Theorems
Chapter IV. GENERAL PROPERTIES OF ORTHOGONAL POLYNOMIALS
1. Foundations
2. Roots of Orthogonal Polynomials. A Recursion Formula
3. Relation to the Theory of Continued Fractions
4. The CHRISTOFFEL-DARBOUX Formula. Convergence of Orthogonal Expansions.
5. Changing the Weight Function 74
Chapter V. THE LEGENDRE POLYNOMIALS
1. The RODRIGUES Formula
2. The Generating Function
3. The LAPLACE Integral
4. Expansions in Terms of LEGENDRE Polynomials
Chapter VI. THE JACOBI POLYNOMIALS
1. Generalization of the RODRIGUES Formula
2. The Recursion Formula, Generating Function, and Differential Equation
3. Estimates of the JACOBI Polynomials. The Expansion Problem.
4. TCHEBYSHEFF Polynonmals of the Second Kind
5. The JACOBI Polynomials for a = !, = -!
Chapter VII. THE MOMENT PROBLEM FOR FINITE INTERVALS
1. Formulation of the Problem
2. The Theorems of F. HAUSDORFF
3. Linear Functionals in C and L2
4. Positive Definite Sequences
Chapter VIII. INFINITE INTERVALS
1. Introduction
2. The LAGUERRE Polynomials
3. The Generalized LAGUERRE Polynomials
4. The HERMITE Polynomials
5. The Moment Problem for Infinite Intervals
6. The FAVARD Theorem
Bibliography
Index
Index of Symbols

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