ON THE CONVERGENCE OF SPLINE PRODUCT QUADRATURES FOR CAUCHY PRINCIPAL VALUE INTEGRALS

In 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 - lambda) dx, based on cubic spline interpolation of f, and obtained convergence results for functions f is-an-element-of C(k)[ -, 1], k = 1, 2 or 3. In this report, the same rules are considered and their convergence is investigated for a larger class of function f. An error bound and some uniform convergence results are established, in the case of equally spaced quadrature nodes, for functions f, satisfying a Holder condition of order mu on [ -1,1], 0 < mu less-than-or-equal-to 1.