Minimal cubature formulas exact for Haar polynomials

Abstract

The cubature formulas we consider are exact for spaces of Haar polynomials in one or two variables. Among all cubature formulas, being exact for the same class of Haar polynomials, those with a minimal number of nodes are of special interest. We outline here the research and construction of such cubature formulas.The problem of constructing and analyzing cubature formulas, which integrate exactly a given collection of functions, has been mainly considered before in the cases when these functions are algebraic or trigonometric polynomials (see, for instance, [11,12]). The approximate integration formulas, exact for finite Haar sums, can be found in the monograph [13] and articles [1,2]. Nevertheless, in these works the authors did not consider the question of minimizing the number of nodes while preserving exactness of the formula on such sums. We will use a notion of the Haar polynomials [9,5] and the corresponding definition of the exactness of approximate integration formulas on the mentioned polynomials. It allows us to introduce a notion of a minimal approximate integration formula, exact for the Haar polynomials. The research and construction of such quadrature and cubature formulas was established in [9,10]. In the present article we will outline the main results of these works. Let us mention that in this article we only consider the one-and two-dimensional cases.

On Error Estimates in Sp for Cubature Formulas Exact for Haar Polynomials

“…The problem of constructing cubature formulas possessing the Haar d-property, i.e., formulas exact for Haar polynomials of degree at most d, was solved in the two-dimensional case in [11][12][13][14][15] under the condition that the weight function g(x 1 , x 2 ) ≡ 1. The error estimates for these cubature formulas was derived in [16].…”