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 \[ ∫ 0 ∞ e − x K ( x , y ) f ( x ) d x , \int _0^\infty {{e^{ - x}}K(x,y)f(x)dx,} \] with f ∈ C p [ 0 , ∞ ) ∩ C q ( 0 , ∞ ) , q ≥ p ≥ 0 f \in {C^p}[0,\infty ) \cap {C^q}(0,\infty ),q \geq p \geq 0 , and x i f ( p + i ) ( x ) ∈ C [ 0 , ∞ ) , i = 1 , … , q − p {x^i}{f^{(p + i)}}(x) \in C[0,\infty ),i = 1, \ldots ,q - p , when q > p q > p . They appear for instance in certain Wiener-Hopf integral equations and are of interest if one wants to solve these by a Nyström 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.