Some topics about my PhD thesis: Structure and properties of shift invariant spaces on locally compact abelian groups

PATERNOSTRO VICTORIA

Exposición; NuHAG presentation; 2011

In this thesis we study shift invariant spaces in the context of locally compact abelian (LCA) groups.For $G$ an LCA groups and $Hsubseteq G$ a closed subgroup of $G$ we introduce the notion of {it $H$-invariant space}  or {it shift invariant space under translations in $H$}.  In case when $H$ is a countable discrete subgroup of $G$, we show that the concept of range functions and the techniques of fiberization are valid in this context. Combining these tools, we provide a characterization for $H$-invariant spaces in terms of the fibers of its elements. As a consequence, we prove characterizations of frames and Riesz bases of these spaces extending previous results that were known for the classical case of $R^d$ and the lattice $Z^d$.On the other hand, we study the problem of {it extra invariance} of $H$-invariant spaces.Our results of extra invariance state several necessary and sufficient conditions for an $H$-invariant spaces  to be invariant along translations in a closed subgroup of $G$, $M$,  containing $H$. In addition we  show that for each closed subgroup $M$ of $G$ which contains $H$ there exists  an $H$-invariant space $V$ that is exactly $M$-invariant. That is, $V$ is not invariant under any other subgroup $M´$ containing $M$.We also obtain estimates on the support of the Fourier transform of the generators of the $H$-invariant spaces, related to its $M$-invariance.  Lastly, we  investigate the structure of those closed subspace of $L^2(G)$ which are invariant by translations along $K$ and also invariant under modulations in $Lambda$, begin $K$ and $Lambda$ closed subgroups of $G$ and the dual group of $G$ respectively. We obtain a characterization of these spaces when $K$ and $Lambda$ are discrete.