Nonlinear approximation with redundant dictionaries

Redundant function dictionaries allow for a multitude of data decompositions. While this adds flexibility one also have to search for the `best representations' adding complexity to the algorithms. It is thus important to know that a redundant function dictionary can represent a given function more efficiently than a classical orthonormal basis. For this purpose, it is important to characterize the approximation spaces associated with the dictionary. The approximation spaces give information on the optimal performance (or potential) of the dictionary. We study so-called Jackson and Bernstein inequalities for general redundant dictionaries. The inequalities give information on the 'size' of the approximation spaces.