  In this paper we are interested in the inverse problem of recovering a
compact supported function from its truncated Fourier transform. We derive new
Lipschitz stability estimates for the inversion in terms of the truncation
parameter. The obtained results show that the Lipschitz constant is of order
one when the truncation parameter is larger than the spatial frequency of the
function, and it grows exponentially when the truncation parameter tends to
zero. Finally, we present some numerical examples of reconstruction of a
compactly supported function from its noisy truncated Fourier transform. The
numerical illustrations validate our theoretical results.
