We obtain new Kolmogorov-type sharp inequalities estimating the norm of the Marchaud fractional derivative \( {\left\Vert {D}_{-}^kf\right\Vert}_{\infty } \) of a function f defined on the positive half line in terms of ‖f‖_p, 1 < p < ∞ , and ‖f〞‖₁. We also solve the following related problems: the Stechkin problem of the best approximation of the operator \( {D}_{-}^k \) by linear bounded operators and the problem of the best possible recovery of the operator \( {D}_{-}^k \) on a class of elements given with errors.

中文翻译：

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正半线上定义的函数的分数导数范数的Kolmogorov型不等式

我们获得新的Kolmogorov型尖锐不等式估计的Marchaud分数阶导数常态\（{\左\ Vert的{d} _ { - } ^ KF \右\ Vert的} _ {\ infty} \）的函数的˚F上定义正半线，用“ f ” _p，1 <  p  <∞和“ f〞” _1表示。我们还解决了以下相关问题：线性有界算子对算子\（{D} _ {-} ^ k \）的最佳逼近的Stechkin问题以及算子\（{D } _ {-} ^ k \）上给出错误的元素类。