The spectral function of shift-invariant spaces
Let \( V \) be a shift-invariant closed subspace of \( L^2 (\mathbb R^n)\) generated by \(\Phi \subset L^2 (\mathbb R^n )\), \[ S(\Phi ) = \overline{\text{span}} \{\varphi(\cdot - k) : \varphi \in \Phi, \; k \in\mathbb Z^n\}. \] If \(\{ \varphi(\cdot - k) : \varphi \in \Phi , k \in \mathbb Z^n\}\) forms a tight frame of \( V= S(\Phi ) \) with constant 1, then the spectral function \( \sigma_\nu \) is given by \[ \sigma_\nu (\xi) = \sum_{\varphi \in \Phi } \; |\widehat \varphi (\xi )|^2. \] In the paper, the spectral function is systematically studied. It is applied to characterize the approximation order of \( S(\Phi )\), and to give a complete characterization of dimension functions. Further, an analogue of the Calderón reproducing formula for generalized MRAs is given and an elementary proof of Rieffel's incompleteness theorem for Garbor systems is presented.
scientific article; zbMATH DE number 1997320
scientific article