THE APPROXIMATION OF MULTIPLE INTEGRALS BY USING INTERPOLATORY CUBATURE FORMULAE

Abstract

This chapter discusses the approximation of multiple integrals by using interpolatory cubature formula. It presents a survey of the theory of interpolatory cubature formulae that has been developed after 1970. In the chapter the following are considered: lower bounds for the number of knots of cubature formula, which are exact for polynomials of some fixed degree, the connection between orthogonal polynomials and cubatureformulae, the method of reproducing kernels, and invariant formula. The extension of cubature formulae of Gaussian type for the multivariate case is important in the theory of interpolatory cubature formula. If two polynomials of degree k orthogonal with respect to Ω and weight-function p (x, y) have exactly k2 roots in common, finite and distinct, then these roots can be taken as knots of a cubature formula for an integral on Ω with the weight-function p (x, y). This formula is exact for all polynomials of degree not higher than 2k - 1.