On the stability of sparse convolutions
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- 作者：Philipp Walka ; philipp.walk@tum.de ; Peter Jungb ; peter.jung@tu-berlin.de ; Gö ; tz E. Pfanderc ; g.pfander@jacobs-university.de
- 关键词：Reverse Young inequality ; Discrete convolution ; Sparsity ; Additive combinatorics ; Fourier minors ; Freiman isomorphism ; Polynomial estimations
- 刊名：Applied and Computational Harmonic Analysis
- 出版年：2017
- 出版时间：January 2017
- 年：2017
- 卷：42
- 期：1
- 页码：117-134
- 全文大小：775 K
- 卷排序：42
文摘
We give a stability result for sparse convolutions on ℓ2(G)×ℓ1(G)ℓ2(G)×ℓ1(G) for torsion-free discrete Abelian groups G such as ZZ. 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.