# A Note on Band-limited Minorants of an Euclidean Ball

## TL;DR

This paper investigates the problem of constructing band-limited functions that are bounded above by an indicator function of a Euclidean ball, focusing on the critical support radius for positive integral existence.

## Contribution

It provides a computation of the critical radius of Fourier support for band-limited minorants of Euclidean balls, advancing understanding of the Beurling-Selberg problem in this context.

## Key findings

- Determined the critical radius for positive integral of band-limited minorants.
- Established conditions under which such minorants exist.
- Contributed to the theory of band-limited approximation of indicator functions.

## Abstract

We study the Beurling-Selberg problem of finding band-limited $L^1$-functions that lie below the indicator function of an Euclidean ball. We compute the critical radius of the support of the Fourier transform for which such construction can have a positive integral.