Optimal quadrature problem on Hardy-Sobolev classes

Let H˜∞,βr denote those 2π-periodic, real-valued functions f on R, which are analytic in the strip Sβ≔{z∈C:|Imz|<β}, β>0 and satisfy the restriction |f(r)(z)|⩽1, z∈Sβ. Denote by [x] the integral part of x. We prove that the rectangular formulaQN*(f)=2πN∑j=0N-1f2πjNis optimal for the class of functions H˜∞,βr among all quadrature formulae of the formQ2N(f)=∑i=1n∑j=0νi-1aijf(j)(ti),where the nodes 0⩽t1<⋯