LAUSR

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.