Approximation of derivatives of analytic functions from one Hardy class by another Hardy class

Abstract

In the Hardy space Hp(De), 1 ≤ p ≤ ∞, of functions analytic in the disk De = {z ∈ C: |z| < e}, we denote by NHp(De), N > 0, the class of functions whose Lp-norm on the circle γe = {z ∈ C: |z| = e} does not exceed the number N and by ∂Hp(De) the class consisting of the derivatives of functions from 1Hp(De). We consider the problem of the best approximation of the class ∂Hp(Dρ) by the class NHp(DR), N > 0, with respect to the Lp-norm on the circle γr, 0 < r < ρ < R. The order of the best approximation as N → +∞ is found: (Equation presented) In the case where the parameter N belongs to some sequence of intervals, the exact value of the best approximation and a linear method implementing it are obtained. A similar problem is considered for classes of functions analytic in rings. © 2019 Mofid University - Center for Human Rights Studies. All rights reserved.