FoCM 2014 conference

Workshop A2 - Computational Harmonic Analysis, Image and Signal Processing

No date set

## Finitely generated shift invariant spaces with extra invariance nearest to observed data

### Carolina Mosquera

### Universidad de Buenos Aires, CONICET, Argentina - mosquera@dm.uba.ar

Let $m, \ell\in\mathbb{N},$ $M$ be a closed subgroup of $\mathbb{R}^d$ containing $\mathbb{Z}^d$ and $F= \{f_1, \dots, f_m\}\subset L^2(\mathbb{R}^d).$ We study the problem of finding the shift invariant space $V$ of length less or equal to $\ell$ which is also $M-$invariant such that $V$ is ``closest'' to the functions $F$ in the sense that $$ V= argmin_{V^{\prime}\in V_M^{\ell}} \sum_{j=1}^m \|f_j- P_{V^{\prime}}f_j\|^2, $$ where $V_M^{\ell}$ is the set of all shift invariant spaces $V^{\prime}$ of length less or equal to $\ell$ which are also $M-$invariant, and $P_{V^{\prime}}$ is the orthogonal projection on $V^{\prime}.$

Also we consider this problem for a particular set of translation invariant spaces.

Joint work with Carlos Cabrelli (Universidad de Buenos Aires, CONICET, Argentina).