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The structure of finitely generated shift-invariant subspaces on locally compact abelian groups. (English) Zbl 1466.43003

Throughout \(G\) will denote a locally compact Abelian group and \(\widehat{G}\) will denote its dual group. Fix a lattice \(L\) in \(G\). A closed subspace \(V\) of \(L^2(G)\) is defined to be shift invariant with respect to \(L\) if \(V\) is invariant under translations by elements in \(L\). For a subset \(\Omega\) of \(L^2(G)\), let \(V(\Omega)\) be the smallest shift invariant subspace of \(L^2(G)\) with respect to \(L\) containing \(\Omega\). The bracket product of \(f, g \in L^2(\widehat{G})\) is given by \[ [f, g](\xi) = \sum_{\eta \in L^{\perp}} f(\xi \eta)\overline{g(\xi\eta)}, \] where \(\xi \in \widehat{G}\). For \(\varphi \in L^2(G)\) define \[ w_{\varphi}(\xi) := [\widehat{\varphi}, \widehat{\varphi} ](\xi ), \] where \(\xi \in \widehat{G}\) and \(\widehat{\varphi}\) is the Fourier transform of \(\varphi\). Denote by \(L^2(\widehat{L}, w_{\varphi})\) the set of all functions \(r:\widehat{L} \rightarrow \mathbb{C}\) that satisfy \[ \int_{\widehat{L}} \vert r(\xi)\vert^2 w_{\varphi}(\xi) \, d\xi < \infty, \] where \(d\xi\) is the Plancherel measure on \(\widehat{L}\). Represent the projection of \(f\) on \(V(\Omega)\) by \(P_{\Omega}(f).\)
The main result of the paper under review is: Let \(\Omega := \{ \varphi_i\}_{i=1}^N \subseteq L^2(G)\) be a finitely minimal generating set for \(V(\Omega)\). Then for each \(f \in V(\Omega)\) we have: \[ \widehat{f} =\sum_{i=1}^N m_i(f) \widehat{\varphi_i},\quad \hbox{where } \quad m_i(f) := \frac{[\widehat{f}, \widehat{h_i}]}{[\widehat{h_i}, \widehat{h_i}]} \in L^2(\widehat{L}, w_{h_i}), \] and \[ \sum_{i=1}^N \Vert m_i (f)\Vert^2_{L^2(\widehat{L}, w_{h_i})} \leq \Vert f \Vert_{L^2(G)}^2, \] where \[ \widehat{h_i} = \widehat{\varphi_i} - \widehat{P_{\Omega^{(i)}} (\varphi_i)}, \] and \(\Omega^{(i)} := \Omega\setminus \{\varphi_i\}\).

MSC:

43A15 \(L^p\)-spaces and other function spaces on groups, semigroups, etc.
43A25 Fourier and Fourier-Stieltjes transforms on locally compact and other abelian groups
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
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