Numerical Algorithms5 (1993)443--452 443 An algorithm for numerical integration based on quasi-interpolating splines C. Dagnino and V. Demichelis Universith di Torino, Dipartirnento di Matematica, Via Carlo Alberto 10, 10123 Torino, ltaly E. Santi Universitgtdi L 'Aquila, Dipartimento di Energetica, 67100 L'Aquila, Italy In this paper product quadratures based on quasi-interpolating splines are proposed for the numerical evaluation of integrals with an Ll-kernel and of Cauchy Principal Value inte- grals. AMS subject classification: 65D30, 65D32. 1. Introduction Splines have been used for numerical integration ever since they entered the numerical analysis scene [13]. However, only recently they have been applied to the numerical evaluation of integrals such as I(Kf) = K(x)f(x)dx, (1) I and of Cauchy principal value integrals such as J(uf;A)=- u(x) dx, -l<A<l, (2) 1 where KeLI[-1, 1],f is bounded in [-1, 1] for case (1) and u andf are such that J(uf; A) exists for case (2). Some authors [1-6,16] have proposed and studied product rules for (1) and (2) based on interpolating or approximating splines. However, their results are not completely satisfactory, since they have some restrictions on the spacing of spline knots [2-6], or on the accuracy of the quadrature [16], or on the convergence prop- erties [1]. ~rWork sponsored by "Ministero dell'Universit~ e Ricerca Scientifica"of Italy. 9 J.C. Baltzer AG, Science Publishers