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

The problem of constructing and analyzing cubature formulas that are exact for a given set of functions was earlier considered primarily as applied to the computation of integrals exact for algebraic and trigonometric polynomials. For example, the approximate integration formulas of algebraic accuracy can be found in [1, 2]. The cubature formulas exact for trigonometric polynomials in particular were studied in [3–7]. The approximate integration formulas exact for the system of Haar functions can be found in the monograph [8]. The accuracy of approximate integration formulas for finite Haar sums was used in [8] to derive error estimates for these formulas. A description of all minimal weighted quadrature formulas possessing the Haar d-property, i.e., formulas exact for Haar functions of groups with indices not exceeding a given number d, was given in [9]. The error estimates for quadrature formulas possessing the Haar d-property in the case of the weight function g(x) ≡ 1 were obtained in [10]. In particular, in the mentioned paper the upper estimate for the norm of the error functional ∥δN∥S∗ p was found for the quadrature formulas having the Haar d-property: