Fourier uniqueness pairs of powers of integers

TL;DR

This paper establishes conditions under which Schwartz functions that vanish at specific powers of integers and their Fourier transforms must be identically zero, contributing to the understanding of Fourier uniqueness pairs and uncertainty principles.

Contribution

It introduces new conditions on parameters for Fourier uniqueness pairs involving powers of integers, extending recent results in uncertainty principles.

Findings

Proves that certain Schwartz functions vanish identically under specified conditions.

Complements recent results in uncertainty principles and Fourier analysis.

Provides new insights into Fourier uniqueness pairs involving powers of integers.

Abstract

We prove, under certain conditions on $(\alpha,\beta)$, that each Schwartz function $f$ such that $f(\pm n^{\alpha}) = \hat{f}(\pm n^{\beta}) = 0, \forall n \ge 0$ must vanish identically, complementing a series of recent results involving uncertainty principles, such as the pointwise interpolation formulas by Radchenko and Viazovska and the Meyer-Guinnand construction of self-dual crystaline measures.