  We employ functional analysis techniques in order to deduce that some
classical and recent interpolation results in Fourier analysis can be suitably
perturbed. As an application of our techniques, we obtain generalizations of
Kadec's 1/4-theorem for interpolation formulae in the Paley-Wiener space both
in the real and complex case, as well as a perturbation result on the recent
Radchenko-Viazovska interpolation result and the
Cohn-Kumar-Miller-Radchenko-Viazovska result for Fourier interpolation with
derivatives in dimensions 8 and 24. We also provide several applications of the
main results and techniques, all relating to recent contributions in
interpolation formulae and uniqueness sets for the Fourier transform.
