On the stability of sparse convolutions

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, that is, injectivity with an...

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Bibliographic Details
Published in:Applied and computational harmonic analysis 2017-01, Vol.42 (1), p.117-134
Main Authors: Walk, Philipp, Jung, Peter, Pfander, Götz E.
Format: Article
Language:English
Subjects:
Additive combinatorics
Discrete convolution
Fourier minors
Freiman isomorphism
Polynomial estimations
Reverse Young inequality
Sparsity
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Summary: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, that is, injectivity 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 Young inequality for sparse convolutions. Our result hinges on a compression argument in additive set theory.
ISSN:1063-5203
1096-603X
DOI:10.1016/j.acha.2015.08.002