On the Stability of Sparse Convolutions

TL;DR

This paper establishes a stability property for sparse convolutions on torsion-free discrete Abelian groups, showing that sparsity prevents cancellation and ensures injectivity with a universal lower bound.

Contribution

It provides the first stability result for sparse convolutions on torsion-free groups, using additive set theory to prevent cancellation effects.

Findings

Sparse convolutions are stable with a universal lower bound.

Torsion-free property prevents full cancellation in convolutions.

The result applies to groups like the integers.

Abstract

We give a stability result for sparse convolutions on $\ell^2(G)\times \ell^1(G)$ for torsion-free discrete Abelian groups $G$ such as $\mathbb{Z}$. It turns out, that the torsion-free property prevents full cancellation in the convolution of sparse sequences and hence allows to establish stability in each entry, that is, for any fixed entry of the convolution the resulting linear map is injective with an universal lower norm bound, which only depends on the support cardinalities of the sequences. This can be seen as a reverse statement of the famous Young inequality for sparse convolutions. Our result hinges on a compression argument in additive set theory.