Interpolatory product quadratures for Cauchy principal value integrals with Freud weights

Abstract

We prove convergence results and error estimates for interpolatory product quadrature formulas for Cauchy principal value integrals on the real line with Freud-type weight functions. The formulas are based on polynomial interpolation at the zeros of orthogonal polynomials associated with the weight function under consideration. As a by-product, we obtain new bounds for the derivative of the functions of the second kind for these weight functions.Mathematics Subject Classification (1991): 41A55; 33C45, 65D30, 65R10

Boundedness and Uniform Numerical Approximation of the Weighted Hilbert Transform on the Real Line

“…Interest in the numerical evaluation of the weighted Hilbert transform is primarily due to the fact that integral equations with Cauchy principal value integrals have shown to be an adequate tool for the modelling of many physical situations. However, only a small number of publications, see [7,13], deal with this problem for the large classes of functions and weights presented here. Typically, our classes of functions are allowed to increase exponentially without bound near AE 1 and so our weights are chosen to counteract this growth.…”

“…The idea of using the sequence fL nþ2 g was merely to obtain the correct order log n which follows by Theorem 1.4. In [7], we used another idea, namely estimates of functions of the second kind although we did not obtain as a good an estimate as log n . We believe that this latter idea will ultimately lead to the correct order log n as well.…”

“…It appears in numerous areas in analysis, in particular, weighted approximation, numerical quadrature, integrable systems, and orthogonal polynomials. We refer the reader to [3,7,8,9,10,11,13,14,16,17,20,21,24,25,28,29,30,31,32,33,37,38] and the many references cited therein for a detailed account of this vast topic. Our interest in the operator given by (1) is to firstly establish its boundedness in suitable weighted subspaces of L 1;w and secondly, to numerically approximate it by a quadrature procedure, which is based on polynomial interpolation at the zeros of orthogonal polynomials associated with the weight w under consideration augmented by two carefully chosen points.…”

“…We briefly mention that the quadrature formulae obtained according to this product integration approach, but using the zeros of the orthogonal polynomials only and not the extra points x 0;n and x nþ1;n ; have been investigated in [7] for a class of admissible Freud weights using estimates for functions of the second kind. Comparing our new results with those of that paper, we see that the introduction of the extra points leads to improved error estimates.…”

An Analytic and Numerical Analysis of Weighted Singular Cauchy Integrals with Exponential Weights on $\mathbb R$

“…for a class of functions f : I → R + for which I[f ; x] is finite. In [1,2,3,4], the first two authors studied this problem and some of its applications for even exponential weights w on (−∞, ∞) of smooth polynomial decay at ±∞ and given regularity. Notice that the argument in the integrand is of the form gw 2 for suitable g : I → R + and weight w 2 .…”