Orthogonal Functions

Moment Theory and Continued Fractions

William Jones, A. Sri Ranga

• Publisher: CRC Press
• Year: 2020
• Print ISBN: 9780824702076
• eBook ISBN: 9781000153675
• Topic: Mathematics
• Subtopic: Mathematics General
• Index: Mathematics

"Oulines an array of recent work on the analytic theory and potential applications of continued fractions, linear functionals, orthogonal functions, moment theory, and integral transforms. Describes links between continued fractions. Pade approximation, special functions, and Gaussian quadrature."

Chebyshev–Laurent Polynomials and Weighted Approximation

ELIANA X. L. DE ANDRADE and DIMITAR K. DIMITROV¹ Departamento de Ciências de Computação e Estatística, lnstituto de Biociências, Letras e Ciências Exatas, Universidade Estadual Paulista, 15054-000 - São José do Rio Preto, SP, Brazil

Let (a, b) ⊂ (0, ∞) and for any positive integer n, let S_n be the Chebyshev space in [a, b] defined by S_n := span { x⁻ⁿ/^2+k, k = 0,…, n }. The unique (up to a constant factor) function τ_n ∈ S_n, which satisfies the orthogonality relation ${\displaystyle {\int }_a^b{\tau }_n\left(x\right)q\left(x\right){\left(x\left(b-x\right)\left(x-a\right)\right)}^{-1/2}dx}=0$ for any q ∈ S_{n – 1}, is said to be the orthogonal Chebyshev S_n-polynomials. This paper is an attempt to exibit some interesting properties of the orthogonal Chebyshev S_n-polynomials and to demonstrate their importance to the problem of approximation by S_n-polynomials. A simple proof of a Jackson-type theorem is given and the Lagrange interpolation problem by functions from S_n is discussed. It is shown also that τ_n obeys an extremal property...

Table of contents

• Cover
• Half Title
• Title Page
• Copyright Page
• Preface
• Table of Contents
• Contributors
• Participants
• 1. Chebyshev-Laurent Polynomials and Weighted Approximation
• 2. Natural Solutions of Indeterminate Strong Stieltjes Moment Problems Derived from PC-Fractions
• 3. A Class of Indeterminate Strong Stieltjes Moment Problems with Discrete Distributions
• 4. Symmetric Orthogonal L-Polynomials in the Complex Plane
• 5. Continued Fractions and Orthogonal Rational Functions
• 6. Interpolation of Nevanlinna Functions by Rationals with Poles on the Ral Line
• 7. Symmetric Orthogonal Laurent Polynomials
• 8. Interpolating Laurent Polynomials
• 9. Computation of the Binet and Gamma Functions by Stieltjes Continued Fractions
• 10. Formulas for the Moments of Some Strong Moment Distributions
• 11. Orthogonal Laurent Polynomials of Jacobi, Hermite, and Laguerre Types
• 12. Regular Strong Hamburger Moment Problems
• 13. Asymptotic Behavior of the Continued Fraction Coefficients of a Class of Stieltjes Transforms Including the Binet Function
• 14. Uniformity and Speed of Convergence of Complex Continued Fractions K(an/1).
• 15. Separation Theorem of Chebyshev–Markov-Stieltjes Type for Laurent Polynomials Orthogonal on (0, ∞)
• 16. Orthogonal Polynomials Associated with a Nondiagonal Sobolev Inner Product with Polynomial Coefficients
• 17. Remarks on Canonical Solutions of Strong Moment Problems
• 18. Sobolev Orthogonality and Properties of the Generalized Laguerre Polynomials
• 19. A Combination of Two Methods in Frequency Analysis: The R(N)-Process
• 20. Zeros of Szegö Polynomials Used in Frequency Analysis
• 21. Some Probabilistic Remarks on the Boundary Version of Worpitzky’s Theorem