### 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.

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