Approximation of Derivatives of Analytic Functions from One Hardy Class by Another Hardy Class

Abstract

In the Hardy space Hp(Dϱ), 1 ≤ p ⪯ ∞, of functions analytic in the disk Dϱ = {z ∈ ℂ}: z < ϱ, we denote by NHp(Dϱ), N > 0, the class of functions whose Lp-norm on the circle γϱ = {z ∈ ℂ: z = ϱ} does not exceed the number N and by ∂Hp(Dϱ) the class consisting of the derivatives of functions from 1Hp(Dϱ). 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: $$\varepsilon (\partial {H^p}({D_\rho }),N{H^p}){)_{{L^p}({\Gamma _r})}} \asymp {N^{ - \beta }}^{/\alpha }{\ln ^{1/\alpha }}N,\alpha = \frac{{\ln R - \ln \rho }}{{\ln R - \ln r}},\beta = 1 - \alpha.$$ 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 annuli.