On bivariate Gaussian cubature formulae

Abstract

Abstract. It is shown that for two classes of integrals the results of Gaussian quadrature can be extended straightforwardly to the bivariate case. For these classes Gaussian formulae of an arbitrary degree are derived.

Minimal cubature rules and polynomial interpolation in two variables

“…The orthogonal polynomials with respect to W α,β,γ were first studied by Koornwinder in [6] and further studied in [7,8,14]. They were applied to study cubature rules in [13]. In the case of γ = ± 1 2 , the orthogonal polynomials with respect to W ± 1 2 can be given explicitly.…”

“…We shall show that these rules can be transformed into minimal cubature rules for W α,β,−1/2 on [−1, 1] 2 in the next section. The first proof that Gaussian cubature rules exist for W ± 1 2 was given in [13] via the structure matrices of orthogonal polynomials. Below is another proof that is of independent interest.…”

“…At the moment, they are known to exist only in two cases. The first case, discovered in [13], is for a family of weight functions that includes, in particular, W α,β,±12 defined by…”

C. F. Dunkl and Y. Xu, Orthogonal Polynomials of Several Variables

“…In Mysovskikh and Chernitsina [1971], a Gaussian cubature formula of degree 5 is constructed numerically, which answers the original question of Radon. One natural question is whether there exists a weight function for which Gaussian cubature formulae exist for all n. The question is answered only recently in Berens et al [1995b] and Schmid and Xu [1994], where two families of weight functions that admit Gaussian cubature formulae are given for d = 2 in Schmid and Xu [1994] and d ≥ 2 in Berens et al [1995b]. We describe the results below…”

A class of bivariate orthogonal polynomials and cubature formula

“…Likewise, if L = dα, then we refer to measure α instead of L . The examples of Gaussian cubature for all n are found only very recently: in [15] we showed that there are actually many Gaussian cubatures; two classes of weight functions are given that admit such formulae. To describe the result, let α be a nondecreasing function on R with finite moments and infinite support.…”