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On Error Estimates for Weighted Quadrature Formulas Exact for Haar Polynomials

The problem of constructing and analyzing cubature formulas that integrate a given collection of functions exactly has been earlier considered mainly in the cases when these functions are algebraic or trigonometric polynomials. The approximate integration formulas exact for finite Haar sums can be found in the monograph [1], the accuracy of approximate integration formulas for finite Haar sums was used there for deriving the error of these formulas. A description of all minimal weighted quadrature formulas possessing the Haar d–property, i.e., formulas exact for Haar polynomials of degree at most d, was given in [2]. In [3] the estimates of the norm of the error functional for quadrature formulas exact for Haar polynomials were proved on the spaces Sp and Hα in the case of the weight function g(x) ≡ 1. In [4] the estimates of the norm of the error functional for minimal quadrature formulas exact for Haar polynomials were obtained on the spaces Sp in the case of the weight function g ∈ L∞[0, 1]. In the two-dimensional case, the problem of constructing cubature formulas possessing the Haar d-property was considered in [5–9]. Estimates of the norm of the error functional for cubature formulas on the spaces Sp and Hα were obtained in [10,11]. The results obtained in [3] are extended in this paper to the case of weighted quadrature formulas on the spaces Sp. In the case of the weight function g ∈ L1[0, 1], a lower estimate of the norm of the error functional δN is derived for quadrature formulas exact for any constant. In the cases of the weight function g ∈ L∞[0, 1] and g ∈ Lq[0, 1] (p −1 + q = 1), upper estimates of the norm of the error functional δN are proved for quadrature formulas possessing the Haar d-property. It is shown that in the case of N ≍ 2 with d → ∞, the value ‖δN‖S∗ p for the formulas under study has the best convergence rate to zero, which is equal to N 1 p with N → ∞.