An error analysis for numerical multiple integration. II

Abstract

tions. This paper describes the application of the hypercircle inequality [12] to the problem of numerical multiple integration and the resulting cubatures. A convergence theorem is given for the cubatures, as well as numerical examples for some particular functions integrated over a square. The cubatures can be compared with what the author has called minimum norm cubatures [5]. New results are given in this paper concerning asymptotic properties of the minimum norm cubatures and these, in turn, imply results concerning the cubatures. 2. Derivation of the Optimal Cubatures. The hypercircle inequality was originally described by Synge [18] and reformulated by Golomb and Weinberger [12] and Davis [10]. It has been used in quadrature theory for analytic functions by Valentin [19] and the author [2] and for functions with integrable nth derivative by Secrest [15], [16]. In the latter cases, this method leads to the theory of spline approximation, which has a large literature to which references can be found in Secrest's articles and in the paper by Birkhoff and de Boor [7]. The cubatures derived from the hypercircle inequality are optimal in a certain sense, and hence this name. Suppose that we want to compute L(f) = Jj f, for some given function f, where the integral is over the square 0 < x, u < 1, and that the following two types of information are available: