The basis functions of Fourier interpolation

TL;DR

This paper investigates the properties of basis functions used in Fourier interpolation, providing size estimates, zero distribution analysis, and implications for Fourier uniqueness, revealing complex behaviors and limitations of these functions.

Contribution

The study offers a detailed analysis of the basis functions' size, zero distribution, and their failure to form a Riesz basis, advancing understanding of Fourier interpolation's mathematical structure.

Findings

Derived asymptotic estimates for zeros near the real line

Discovered existence of Fourier nonuniqueness pairs with arbitrarily large excess

Showed basis functions do not form a Riesz basis in the considered Hilbert space

Abstract

The basis functions of the Fourier interpolation formula of Radchenko and Viazovska, constructed by means of weakly holomorphic modular forms for the Hecke theta group, are entire functions of order $2$ having interesting time-frequency properties. We give precise size estimates and study the distribution of zeros of these functions. We give in particular asymptotic estimates for the location and the number of extraneous zeros on or close to the real line. This result reveals the surprising existence of Fourier nonuniqueness pairs whose apparent ``excess'' compared to the Fourier uniqueness pair of Radchenko and Viazovska may be made arbitrarily large. Our estimates also show that the basis functions fail to yield a Riesz basis in the Hilbert space used by Kulikov, Nazarov, and Sodin in their recent study of Fourier uniqueness pairs. Some numerical data are presented, suggesting additional fine scale properties.