Convergence of product integration rules over $(0,\infty )$ for functions with weak singularities at the origin

Abstract:

In this paper we consider integrals of the form \[ \int _0^\infty {{e^{ - x}}K(x,y)f(x)dx,} \] with $f \in {C^p}[0,\infty ) \cap {C^q}(0,\infty ),q \geq p \geq 0$, and ${x^i}{f^{(p + i)}}(x) \in C[0,\infty ),i = 1, \ldots ,q - p$, 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 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.