On the simultaneous approximation of a Hilbert transform and its derivatives on the real semiaxis

Abstract

In this paper we propose a global method to approximate the derivatives of the weighted Hilbert transform of a given function fHp(fwα,t)=dpdtp⨍0+∞f(x)x−twα(x)dx=p!⨎0+∞f(x)(x−t)p+1wα(x)dx, where p∈{1,2,…}, t>0, and wα(x)=e−xxα is a Laguerre weight. The right-hand integral is defined as the finite part in the Hadamard sense. The proposed numerical approach is convenient when the approximation of the function Hp(fwα,t) is required. Moreover, if there is the need, all the computations can be performed without differentiating the density function f. Numerical stability and convergence are proved in suitable weighted uniform spaces and numerical tests which confirm the theoretical estimates are presented.