# Uncertainty Principles for Fourier Multipliers

## TL;DR

This paper investigates the Sobolev regularity constraints for functions with zeros on the torus whose reciprocals act as Fourier multipliers, connecting these properties to Gabor systems and extending the Balian-Low uncertainty principle.

## Contribution

It introduces new bounds on Sobolev regularity for Fourier multipliers with zeros, explores matrix-valued and non-symmetric cases, and links these to Gabor system approximation properties.

## Key findings

- Quantified Sobolev regularity for functions with zeros on the torus.
- Extended Balian-Low uncertainty principle in new settings.
- Connected Fourier multiplier properties to Gabor system approximations.

## Abstract

The admittable Sobolev regularity is quantified for a function, $w$, which has a zero in the $d$--dimensional torus and whose reciprocal $u=1/w$ is a $(p,q)$--multiplier. Several aspects of this problem are addressed, including zero--sets of positive Hausdorff dimension, matrix valued Fourier multipliers, and non--symmetric versions of Sobolev regularity. Additionally, we make a connection between Fourier multipliers and approximation properties of Gabor systems and shift--invariant systems. We exploit this connection and the results on Fourier multipliers to refine and extend versions of the Balian--Low uncertainty principle in these settings.