Assume that we need to process a huge amount of data from which we know that it is low dimensional, but has possibly been corrupted by noise. A common practice in this case, is to impose some hypothesis on the data and use a standard model. For example we can assume that the data is band limited, then we can use an appropriate Paley-Wiener space. A more realistic approach is to select a big class of models and try to find the one that best fits our data.

To deal with noisy data we have recently considered approximation by shift invariant spaces. In this talk we review these results and in particular describe new work for subspaces that include rotations and translations which is very desirable in applications to signal processing.