Perturbed interpolation formulae and applications

Abstract

We employ functional analysis techniques in order to deduce some versions of classical and recent interpolation results in Fourier analysis with perturbed nodes. As an application of our techniques, we obtain generalizations of Kadec’s $\frac{1}{4}$-theorem for interpolation formulae in the Paley–Wiener space both in the real and complex cases, as well as versions of the recent interpolation result of Radchenko and Viazovska (Publ. Math. Inst. Hautes Études Sci. 129 (2019), 51–81) and the result of Cohn, Kumar, Miller, Radchenko and Viazovska (Ann. Math $(2)$ 196:3 (2022), 983–1082) for Fourier interpolation with derivatives in dimensions $8$ and $24$ with suitable perturbations of the interpolation nodes. We also provide several applications of the main results and techniques, relating to recent contributions in interpolation formulae and uniqueness sets for the Fourier transform.