On the best methods for recovering derivatives in Sobolev classes

We construct the best (optimal) methods for recovering derivatives of functions in generalized Sobolev classes of functions on $\mathbb R^d$ provided that for every such function we know (exactly or approximately) its Fourier transform on an arbitrary measurable set $A\subset\mathbb R^d$. In both cases we construct families of optimal methods. These methods use only part of the information about the Fourier transform, and this part is subject to some filtration. We consider the problem of finding the best set for the recovery of a given derivative among all sets of a fixed measure.