On the convergence of spline product quadratures for Cauchy principal value integrals

AbstractIn a recent paper (this journal (1990)), the authors proposed product integration formulas, for the numerical evaluation of the Cauchy principal value integral f1−1u(x) f;(x)/(x−λ) dx, based on cubic spline interpolation of f;, and obtained convergence results for functions f; ϵ Ck[−1, 1], k = 1, 2 or 3. In this report, the same rules are considered and their convergence is investigated for a larger class of functions f;. An error bound and some uniform convergence results are established, in the case of equally spaced quadrature nodes, for functions f;, satisfying a Hölder condition of order μ on [−1, 1], 0 < μ ⩽ 1