  In this communication, we address the problem of approximating the atoms of a
parametric dictionary, commonly encountered in the context of sparse
representations in "continuous" dictionaries. We focus on the case of
translation-invariant dictionaries, where the inner product between atoms only
depends on the difference between parameters. We investigate the following
general question: is there some low-rank approximation of the dictionary $
which interpolates a subset of atoms while preserving the translation-invariant
nature of the original dictionary? We derive necessary and sufficient
conditions characterizing the existence of such an "interpolating" and
"translation-invariant" low-rank approximation. Moreover, we provide
closed-form expressions of such a dictionary when it exists. We illustrate the
applicability of our results in the case of a two-dimensional isotropic
Gaussian dictionary. We show that, in this particular setup, the proposed
approximation framework outperforms standard Taylor approximation.
