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In the class of functions analytic in the annulus \(C_r:=\left\{z\in\mathbb{C}\, :\, r<|z|<1\right\}\) with bounded \(L^p\)-norms on the unit circle, we study the problem of the best approximation of the operator taking the nontangential limit boundary values of a function on the circle \(\Gamma_r\) of radius \(r\) to values of the derivative of the function on the circle \(\Gamma_\rho\) of radius \(\rho,\, r<\rho<1,\) by bounded linear operators from \(L^p(\Gamma_r)\) to \(L^p(\Gamma_ \rho)\) with norms not exceeding a number \(N\).  A solution of the problem has been obtained in the case when \(N\) belongs to the union of a sequence of intervals. The related problem of optimal recovery of the derivative of a function from boundary values of the function on \(\Gamma_\rho\) given with an error has been solved.

APPROXIMATION OF THE DIFFERENTIATION OPERATOR ON THE CLASS OF FUNCTIONS ANALYTIC IN AN ANNULUS