Properties of Interpolatory Product Integration Rules

Abstract

This paper is concerned with the approximate evaluation of $\int_{ - 1}^1 {k(x)f(x)d_x } $, where k is Lebesgue integrable and f is at least Riemann integrable, and preferably smooth. The integral is approximated by a rule of the form $\sum\nolimits_{i = 1}^n {w_{ni} (k)f(x_{ni} )} $, where the points $x_{ni} $ are chosen in some prescribed way, and the weights $w_{ni} (k)$ are such that the rule is exact if f is any polynomial of degree $ < n$. For suitable choices of the points and suitable functions k, the rule is shown to converge to the exact integral, and the companion rule $\sum {|w_{ni} (k)|f(x_{ni} )} $ to$\int {|k(x)|f(x)dx} $, for all Riemann integrable functions f. (The companion property ensures, for example, that the weights have the asymptotic positivity property if k is positive.) Error bounds are given that guarantee rapid convergence of the rule if f is smooth. Finally, by establishing a connection with mean convergence of Lagrangian interpolation some negative theorems are proved. In particular, it is shown that for every choice of the points $\{ x_{ni} \} $ there exist $k \in L_1 $ and $f \in C$ such that the quadrature rule diverges.