Product integration of piecewise continuous integrands based on cubic spline interpolation at equally spaced nodes (Q1093312)

This paper is concerned with the approximate evaluation of \(\int ^{b}_{a}K(x)f(x)dx\), where K(x) is fixed Lebesgue integrable function, by product formulas of the form \(\sum ^{n}_{i=0}w_ if(x_ i)\) based on cubic spline interpolation of the function f. Generally, whenever it is possible, product quadratures incorporate the bad behaviour of the integrand in the kernel K. Here, however, f can have a finite number of jump discontinuities in [a,b]. Convergence results are established and some numerical applications are given for a logarithmic singularity structure in the kernel.