Uniqueness theorems for Fourier transforms

Let Gamma be a smooth curve in the plane R(2). and A be any subset of R(2). When can one recover uniquely a finite measure mu, supported by Gamma and absolutely continuous with respect to the arc length measure on Gamma, from the restriction to Lambda of its Fourier transform? In this note we present two results in the subject, one is concerned with the case when Gamma is a circle, and the other with the case when Lambda is "close" to a lattice. (C) 2010 Elsevier Masson SAS. All rights reserved.