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

In this paper we propose a global method to approximate the derivatives of the weighted Hilbert transform H_0(f w_α, t) of a given function f, where w_α is a Laguerre weight, on the real semiaxis. The proposed numerical approach is convenient when the approximation of the function H_p(f w_α, 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.