Optimal quadrature formulas for classes of functions with bounded mixed difference (Q1174032)

Let \(H^ r_{p,\ell}\) be the class of \(d\)-places periodic functions with the domain of periodicity \(\Omega=[0,1]^ d\), satisfying the conditions \(\| f(x)\|_{L_ p(\Omega)}\leq 1\), \(\|\Delta^ \ell_ hf(x)\|_{L_ p}\leq\prod^ d_{j=1}| h_ j|^{r_ j}\), \(r=(r_ 1,\dots,r_ d)\), \(h=(h_ 1,\dots,h_ d)\), being the mixed difference of \(\ell\)th order. In the present paper the author gives optimal quadratures for such functions.