Convergence of Product Integration Rules Over (0, ∞) for Functions with Weak Singularities at the Origin

Abstract

In this paper we consider integrals of the form ∫ 0 ∞ e −x K(x, y) f(x) dx, with f ∈ C p [0, ∞) ∩ C q (0, ∞), q ≥ p ≥ 0, and x i f (p+1) (x) ∈ C[0, ∞), i = 1,..., q − p, when q > p. They appear for instance in certain Wiener-Hopf integral equations and are of interest if one wants to solve these by a Nystrom method. To discretize the integral above, we propose to use a product rule of interpolatory type based on the zeros of Laguerre polynomials. For this rule we derive (weighted) uniform convergence estimates and present some numerical examples