Error bounds for spline-based quadrature methods for strongly singular integrals (Q1128072)

This paper concerns with the approximation of Hadamard finite-part integrals \(H_p[f]\) with singularities of order \(p\geq 1\) on an interval \(a\leq t\leq b\). The numerical calculation of \(H_p[f]\) is performed by considering methods based on spline approximation for \(f\). The main result states that, if \(d\) denotes the degree of exactness of the approximation process and \(f^{(s)}\) is bounded for \(p-1< s\leq d+1\), then the corresponding error bounds hold uniformly on \([a,b]\). Previous bounds given for some types of spline approximations are improved and generalized. Possible applications and extensions of the results established are mentioned at the end of the paper.