Approximation of the Hilbert Transform on the real line using Hermite zeros

Abstract

Abstract. The authors study the Hilbert Transform on the real line. They introduce some polynomial approximations and some algorithms for its numerical evaluation. Error estimates in uniform norm are given.

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

“…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.…”

“…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.…”

Multi-domain spectral approach for the Hilbert transform on the real line

“…The method, together with rigorous error analysis, is illustrated for several examples in [27]. There are many other numerical approaches for the computation of the Hilbert transform, see for instance [17] for a recent review and [4,7,21,29] for new developments. Some of these approaches compute the Hilbert transform in terms of certain transcendental functions which then have to be computed as well.…”

Multi-domain spectral approach for the Hilbert transform on the real line

“…The method, together with rigorous error analysis, is illustrated for several examples in [27]. There are many other numerical approaches for the computation of the Hilbert transform, see for instance [17] for a recent review and [4,5,21,29] for new developments. Some of these approaches compute the Hilbert transform in terms of certain transcendental functions which then have to be computed as well.…”