Fourier interpolation on the real line

Abstract

In this paper we construct an explicit interpolation formula for Schwartz functions on the real line. The formula expresses the value of a function at any given point in terms of the values of the function and its Fourier transform on the set $\{0, \pm\sqrt{1}, \pm\sqrt{2}, \pm\sqrt{3},\dots\}$ . The functions in the interpolating basis are constructed in a closed form as an integral transform of weakly holomorphic modular forms for the theta subgroup of the modular group.