Error Bounds for a Gauss-Type Quadrature Rule to Evaluate Hypersingular Integrals

Abstract

In the present paper we consider hypersingular integral of the following type
\int_0^\infty f (x)/(x-t)^{p+1}w(x)dx;     (1)
where the integral is understood in the Hadamard finite part sense, p is a positive integer, w(x) = e^{-x}x^\alpha is
a Laguerre weight of parameter \ge 0 and t > 0.

In [6] we proposed an efficient numerical algorithm for
approximating (1), focusing our attention on the computational aspects and on the efficient implementation
of the method. Here, we introduce the method discussing the theoretical aspects, by proving the stability
and the convergence of the procedure for density functions f s.t. f^(p) satisfies a Dini- type condition. For
the sake of completeness, we present some numerical tests which support the theoretical estimates.