Approximation of hypersingular integral transforms on the real axis

where wα,β(x) = |x|αe−|x| β is a generalized Freud weight with α ≥ 0, β > 1 and 0 ≤ p ∈ N. This topic is of interest, for instance, in the numerical solution of hypersingular integral equations, which are often models for physics and engineering problems (see [5, 2, 4]). To our knowledge, most of the papers available in the literature deal with the approximation of Hadamard integrals on bounded intervals (see for instance [4] and the references therein) and the case on the real semiaxis has been considered recently in [1, 3]. We propose here different procedures, which are differently convenient, according that the computation of the integral is required in “many” or “few” values of the parameter t. The convergence and stability of the proposed methods are proved and error estimates are given. Some numerical tests are shown in order to compare their performances.