# Balian-Low in shift-invariant spaces with additional invariance

Akram Aldroubi

given at  strobl11 (14.06.11 15:45)
id:  2074
length:  30min
status:  accepted
type:  talk
A Principle Shift-Invariant Spaces (PSIS) are commonly used in numerical analysis and signal/data processing (also liked by Hans Feichtinger who endearingly calls them {\em spline like spaces}). These spaces are by definition closed subspaces of $L^2(\mathbb R)$ that are invariant under integer translations and can be generated by the span a generator $\phi\subset L^2(\R)$
$$\Sf(\phi) \EQ \clspan\set{T_j \phi: \, j \in \Z}.$$ If $\phi\in L^2(\R)$, $\{\phi(\cdot-k)|\ k\in {\mathbb Z}\}$ is a Riesz basis for its generating space $V(\phi)$, and if $V(\phi)$ is also translation-invariant (i.e., invariant under all translations) then $\phi\not\in L^1({\mathbb R})$. Thus the generators of principal-shift invariant spaces that are also translation invariant are not well-localized in space,e.g., the Paley-Wiener space generated by the function $\phi=\mathrm {sinc}$. Principle shift-invariant spaces that are also translation invariant are desirable in applications, however, the non-localization of their generators may be a drawback for computational reasons. This leads for the search of PSIS that have well-localized generators and that have an additional $\frac{1}{n}\Z$-invariance for some chosen $n>1$, i.e., an invariance under translations by $\frac{1}{n}$. This raises the following questions: 1) Can we characterize the set of functions that generate PSIS (or FSIS) that are also $\frac{1}{n}\Z$-invariant?; 2) Let $n>1$, can we find well-localized in space and frequency simultaneously and that generate PSIS that that are also $\frac{1}{n}\Z$-invariant?