No 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.

We propose a version of the two-dimensional discrete Haar transform with 2 nodes forming Π 0-grids associated with the triangular partial sums of Fourier-Haar series of a given function. Due to the structure of Π 0-grids, the computation of coefficients of this discrete transform is based on a cubature formula with 2 nodes being exact for Haar polynomials of degree at most , owing to that all the coefficients (1 , 2) 1 , 2 of the constructed transform coincide with the Fourier-Haar coefficients (1 , 2) 1 , 2 for Haar polynomials of degree at most −max{ 1 , 2 } (0 1 + 2 , where). The standard two-dimensional discrete Haar transform with 2 nodes does not possess this property.

On the spaces Sp, an upper and lower estimates for the norm of the error functional cubature formulas possessing the Haar d-property are obtained for the n-dimensional case