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Computing The Hilbert Transform Of The Generalized Laguerre And Hermite Weight Functions
, 2000
"... Explicit formulae are given for the Hilbert transform Z R \Gamma w(t)dt=(t \Gamma x), where w is either the generalized Laguerre weight function w(t) = 0 if t 0, w(t) = t ff e \Gammat if 0 ! t ! 1, and ff ? \Gamma1, x ? 0, or the Hermite weight function w(t) = e \Gammat 2 , \Gamma1 ! t ! ..."
Abstract

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Explicit formulae are given for the Hilbert transform Z R \Gamma w(t)dt=(t \Gamma x), where w is either the generalized Laguerre weight function w(t) = 0 if t 0, w(t) = t ff e \Gammat if 0 ! t ! 1, and ff ? \Gamma1, x ? 0, or the Hermite weight function w(t) = e \Gammat 2 , \Gamma1 ! t ! 1, and \Gamma1 ! x ! 1. Furthermore, numerical methods of evaluation are discussed based on recursion, contour integration and saddlepoint asymptotics, and series expansions. We also study the numerical stability of the threeterm recurrence relation satisfied by the integrals Z R \Gamma n (t; w)w(t)dt=(t \Gamma x), n = 0; 1; 2; : : : , where n ( \Delta ; w) is the generalized Laguerre, resp. the Hermite, polynomial of degree n. AMS subject classification: 65D30, 65D32, 65R10. Key words: Hilbert transform, classical weight functions, computational methods. 1
Algorithm xxx — ORTHPOL: A package of routines for generating orthogonal polynomials and Gausstype quadrature rules

, 1993
"... A collection of subroutines and examples of their uses, as well as the underlying numerical methods, are described for generating orthogonal polynomials relative to arbitrary weight functions. The object of these routines is to produce the coefficients in the threeterm recurrence relation satisfie ..."
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A collection of subroutines and examples of their uses, as well as the underlying numerical methods, are described for generating orthogonal polynomials relative to arbitrary weight functions. The object of these routines is to produce the coefficients in the threeterm recurrence relation satisfied by the orthogonal polynomials. Once these are known, additional data can be generated, such as zeros of orthogonal polynomials and Gausstype quadrature rules, for which routines are also provided.
Local Theory of a Collocation Method for Cauchy Singular Integral Equations on an Interval
, 1997
"... We consider a collocation method for Cauchy singular integral equations on the interval based on weighted Chebyshev polynomials, where the coefficients of the operator are piecewise continuous. Stability conditions are derived using Banach algebra methods, and numerical results are given. ..."
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We consider a collocation method for Cauchy singular integral equations on the interval based on weighted Chebyshev polynomials, where the coefficients of the operator are piecewise continuous. Stability conditions are derived using Banach algebra methods, and numerical results are given.