Best quadrature formulae on Hardy–Sobolev classes

For functions in the Hardy–Sobolev class $H_\infty^r$, which is defined as the set of functions analytic in the unit disc and satisfying $f^{(r)}(z)|\leqslant 1$, we construct best quadrature formulae that use the values of the functions and their derivatives on a given system of points in the interval $(-1,1)$. For the periodic Hardy–Sobolev class $H_{\infty,\beta}^r$, which is defined as the set of $2\pi$-periodic functions analytic in the strip $|\operatorname{Im}z|<\beta$ and satisfying $|f^{(r)}(z)|\leqslant 1$, we prove that the rectangle rule is the best for an equidistant system of points, and we calculate the error in this formula. We construct best quadrature formulae on the class $H_{p,\beta}$, which is defined similarly to $H_{\infty,\beta}$, except that the boundary values of functions are taken in the $L_p$-norm. We also construct an optimal method for recovering functions in $H_p^r$ from the Taylor information $f(0),f'(0),\dots,f^{(n+r-1)}(0)$.