Fourier interpolation on the real line

TL;DR

This paper introduces an explicit interpolation formula for Schwartz functions on the real line using weakly holomorphic modular forms, enabling the evaluation of functions based on their values and Fourier transforms at specific algebraic points.

Contribution

It develops a novel interpolation method for Schwartz functions leveraging modular forms and the Hecke theta group, connecting function values with their Fourier transforms at algebraic points.

Findings

Provides an explicit interpolation formula for Schwartz functions.

Connects function values with Fourier transforms at algebraic points.

Utilizes weakly holomorphic modular forms for the Hecke theta group.

Abstract

We use weakly holomorphic modular forms for the Hecke theta group to 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 $\{0, \pm\sqrt{1}, \pm\sqrt{2}, \pm\sqrt{3},\dots\}$.