Approximation of the Differentiation Operator on the Class of Functions Analytic in an Annulus

Abstract

In the class of functions analytic in the annulus Cr:={z∈C:r<|z|<1} with bounded Lp-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 Γr of radius r to values of the derivative of the function on the circle Γρ of radius ρ,r<ρ<1, by bounded linear operators from Lp(Γr) to Lp(Γρ) 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 Γρ given with an error has been solved.