In the Hardy space H^p(D_ϱ), 1 ≤ p ⪯ ∞, of functions analytic in the disk D_ϱ = {z ∈ ℂ}: z < ϱ, we denote by NH^p(D_ϱ), N > 0, the class of functions whose L^p-norm on the circle γ_ϱ = {z ∈ ℂ: z = ϱ} does not exceed the number N and by ∂H^p(D_ϱ) the class consisting of the derivatives of functions from 1H^p(D_ϱ). We consider the problem of the best approximation of the class ∂H^p(D_ϱ) by the class NH^p(DR)N > 0, with respect to the L^p-norm on the circle γ_r, 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.

中文翻译：

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一个Hardy类对另一类Hardy类的解析函数导数的逼近

在哈代空间ħ ^p（d _ρ），1个≤ p ⪯∞，函数在盘解析d _ρ = { ž ∈ℂ}：Ž < ρ，我们用NH ^p（d _ρ），Ñ > 0，之类的其功能大号^p圆上的范数γ _ρ = { ž ∈ℂ：ž = ρ }不超过数ñ和由∂H ^p（d _ρ）由1 H ^p（D _ϱ）的函数的导数组成的类。我们考虑类的最佳近似的问题∂H ^p（d _ρ）由类NH ^p（DR）ñ > 0，相对于所述大号^p圆上的范数γ _{- [R}，0 < - [R < ρ < [R 。找到最佳近似的阶为N →+∞：$$ \ varepsilon（\ partial {H ^ p}（{D_ \ rho}），N {H ^ p}）{）_ {{L ^ p}（{\ Gamma _r}）}} {-\ beta}} ^ {/ \ alpha} {\ ln ^ {1 / \ alpha}} N，\ alpha = \ frac {{\ ln R-\ ln \ rho}} {{\ ln R-\ ln r}}，\ beta = 1-\ alpha。$$在参数N属于某个间隔序列的情况下，可以获得最佳逼近的精确值和实现它的线性方法。对于环中分析的功能类别，也考虑了类似的问题。