Nonlinear approximation with dictionaries,.. II: Inverse estimates

In this paper we study inverse estimates of the Bernstein type for nonlinear approximation with structured redundant dictionaries in a Banach space. The main results are for separated decomposable dictionaries in Hilbert spaces, which generalize the notion of joint block-diagonal mutually incoherent bases introduced by Donoho and Huo. The Bernstein inequality we obtain has an exponent that does not match that of the corresponding Jackson inequality. This may come from the fact that the class of separated decomposable dictionaries contains some highly redundant dictionaries, as we demonstrate with some examples.