2014/09/24 by Philipp Walk, Peter Jung, Walk, Philipp +3
Computer Science · Mathematics · #FOS: Computer and information sciences #Information Theory (cs.IT) #cs.IT #math.IT
paper · pdf · doi:10.48550/arxiv.1409.6874
submitted to Applied and Computational Harmonic Analysis
arxiv created 2014/11/23 · arxiv updated 2014/11/25
We give a stability result for sparse convolutions on ℓ2(G)× ℓ1(G) for torsion-free discrete Abelian groups G such as 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.