Optimal quadrature problem on Hardy-Sobolev classes (Q2576278)

Let \(\widetilde H^r_{\infty,\beta}\) denote \(2\pi\)-periodic, real valued functions \(f\) on \(R\), which are analytic in the strip \(S_\beta:= \{z\in C:| \text{Im}\,z|<\beta\}\), \(\beta>0\) and satisfy the restriction \(|f^{(r)}(z)|\leq 1\), \(z\in X_\beta\). The authors obtain the optimal quadrature formula for the class of functions \(\widetilde H^r_{\infty, \beta}\) among all quadrature formulae of a certain form. Moreover, they exactly determine the error estimate of the optimal quadrature formula on the class \(\widetilde H^r_{\infty,\beta}\).