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Ideals in the convolution algebra of periodic distributions

Sasane, Amol (2024) Ideals in the convolution algebra of periodic distributions. Journal of Fourier Analysis and Applications, 30 (2). ISSN 1069-5869

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Identification Number: 10.1007/s00041-024-10078-y

Abstract

The ring of periodic distributions on Rd with usual addition of distributions, and with convolution is considered. Via Fourier series expansions, this ring is isomorphic to the ring S′(Zd) of all maps f:Zd→C of at most polynomial growth (that is, there exist a real number M>0 and an integer m≥0 such that |f(n)|≤M(1+|n1|+⋯+|nd|)m for all n=(n1,⋯,nd)∈Zd), with pointwise operations. It is shown that finitely generated ideals in S′(Zd) are principal, and ideal membership is characterised analytically. Calling an ideal in S′(Zd) fixed if there is a common index n∈Zd where each member vanishes, the fixed maximal ideals are described, and it is shown that not all maximal ideals are fixed. It is shown that finitely generated (hence principal) prime ideals in S′(Zd) are fixed maximal ideals. The Krull dimension of S′(Zd) is proved to be infinite, while the weak Krull dimension is shown to be equal to 1.

Item Type: Article
Official URL: https://link.springer.com/journal/41
Additional Information: © 2024 The Author
Divisions: Mathematics
Subjects: Q Science > QA Mathematics
Date Deposited: 12 Mar 2024 09:33
Last Modified: 15 May 2024 18:37
URI: http://eprints.lse.ac.uk/id/eprint/122333

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