Linear deviations of the classes $\widetilde W_p^\alpha$ and approximations in spaces of multipliers

We consider the problem of the asymptotically best linear method of approximation in the metric of $L_s[-\pi,\pi]$ of the set $\widetilde W_p^\alpha(1)$ of periodic functions with a bounded in $L_p[-\pi,\pi]$ fractional derivative, by functions from $\widetilde W_p^\beta(M)$ ,beta >agr, for sufficiently large M, and the problem about the best approximation in $L_s[-\pi,\pi]$ of the operator of differentiation on $\widetilde W_p^\alpha(1)$ by continuous linear operators whose norm (as operators from $L_r[-\pi,\pi]$ into $L_q[-\pi,\pi]$)does not exceed $M$. These problems are reduced to the approximation of an individual element in the space of multipliers, and this allows us to obtain estimates that are exact in the sense of the order.