Characterizations of principal shift-invariant spaces.
The paper considers shift-invariant spaces in \( L^1({\mathbb R})\). Given a compactly supported function \(\phi\), let \[ S(\phi ):= \biggl\{ f : f = \sum_{\alpha \in {\mathbb Z}} c(\alpha ) \phi(\cdot -\alpha ),\;c \in l({\mathbb Z}) \biggr\} \] be the principal shift-invariant space generated by \(\phi\). Then it has been shown by \textit{R.-Q. Jia} [Proc. Am. Math. Soc. 125, 785--793 (1997; Zbl 0860.41021)] that with \[ S_p(\phi) := \biggl\{ f : f = \sum_{\alpha \in {\mathbb Z} } c(\alpha) \phi (\cdot -\alpha),\;c \in l_p ({\mathbb Z}) \biggr\} \] we have for \( 1 < p < \infty \) \[ S_p( \phi ) = S(\phi ) \; \cap \; L_p({\mathbb R} ). \] For \(p = 1\), this assertion is not true. In this paper, shift-invariant spaces in \( L_1({\mathbb R} )\) are characterized and sufficient and necessary conditions are given, under which we still have \( S_1(\phi ) = S(\phi ) \, \cap \, L_1({\mathbb R})\). The investigations use special systems of difference equations and generators \(\phi \) being locally linearly independent over \([0, 1 ]\).
scientific article; zbMATH DE number 1836080