This article is cited in
5 papers
On spectral synthesis on element-wise compact Abelian groups
S. S. Platonov Petrozavodsk State University
Abstract:
Let
$G$ be an arbitrary locally compact Abelian group and let
$C(G)$ be the space of all continuous complex-valued functions on
$G$. A closed linear subspace
$\mathscr H\subseteq C(G)$ is referred to as an invariant subspace if
it is invariant with respect to the shifts
$\tau_y\colon f(x)\mapsto f(xy)$,
$y\in G$. By definition, an invariant subspace
$\mathscr H\subseteq C(G)$ admits strict spectral synthesis if
$\mathscr H$ coincides with the closure
in
$C(G)$ of the linear span of all characters of
$G$ belonging to
$\mathscr H$. We say that strict spectral synthesis holds in the space
$C(G)$ on
$G$ if every invariant subspace
$\mathscr H\subseteq C(G)$ admits strict spectral synthesis. An element
$x$ of a topological group
$G$ is said to be compact if
$x$ is contained in some compact subgroup of
$G$. A group
$G$ is said to be element-wise compact if all elements of
$G$ are compact. The main result of the paper is the proof of the fact that strict spectral synthesis holds in
$C(G)$ for a locally compact Abelian
group
$G$ if and only if
$G$ is element-wise compact.
Bibliography: 14 titles.
Keywords:
spectral synthesis, locally compact Abelian groups, element-wise compact groups, Fourier transform on groups, Bruhat-Schwartz functions.
UDC:
517.986.62
MSC: 43A25 Received: 01.09.2014
DOI:
10.4213/sm8419
Fulltext:
PDF file (599 kB)
