Exactness and optimality of methods for recovering functions from their spectrum

Optimal methods are constructed for recovering functions and their derivatives in a Sobolev class of functions on the line from exactly or approximately defined Fourier transforms of these functions on an arbitrary measurable set. The methods are exact on certain subspaces of entire functions. Optimal recovery methods are also constructed for wider function classes obtained as the sum of the original Sobolev class and a subspace of entire functions.