Ideals in the Convolution Algebra of Periodic Distributions

Abstract

The ring of periodic distributions on \(\mathbb {R}^{\texttt {d}}\) with usual addition of distributions, and with convolution is considered. Via Fourier series expansions, this ring is isomorphic to the ring \(\mathcal {S}'(\mathbb {Z}^{\texttt {d}})\) of all maps \(f:\mathbb {Z}^{\texttt {d}}\rightarrow \mathbb {C}\) of at most polynomial growth (that is, there exist a real number \(M>0\) and an integer \(\texttt {m}\ge 0\) such that \( |f(\varvec{n})|\le M(1+|\texttt{n}_1|+\cdots +|\texttt {n}_{\texttt {d}}|)^{\texttt {m}}\) for all \(\varvec{n}=(\texttt{n}_1,\cdots , \texttt {n}_{\texttt {d}})\in \mathbb {Z}^{\texttt {d}}\)), with pointwise operations. It is shown that finitely generated ideals in \(\mathcal {S}'(\mathbb {Z}^{\texttt {d}})\) are principal, and ideal membership is characterised analytically. Calling an ideal in \(\mathcal {S}'(\mathbb {Z}^\texttt{d})\) fixed if there is a common index \(\varvec{n}\in \mathbb {Z}^{\texttt {d}}\) 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 \(\mathcal {S}'(\mathbb {Z}^{\texttt {d}})\) are fixed maximal ideals. The Krull dimension of \(\mathcal {S}'(\mathbb {Z}^{\texttt {d}})\) is proved to be infinite, while the weak Krull dimension is shown to be equal to 1.