  We study the asymptotics near the origin of the Fourier transform in weighted
Hardy spaces of analytic functions in the upper half-plane, and of the Laplace
transform in weighted spaces of entire functions of zero exponential type.
  These results are applied to two closely related problems posed by E.
Dyn'kin: we find the asymptotics of the depth of zero in non-quasianalytic
Denjoy-Carleman classes, and of the exact Levinson-Sjoberg majorant.
