Lower bounds for -term approximations in the metric of the discrete space

Abstract

In applied problems m-term approximations are used extensively. The approximating m-term polynomials are usually constructed by means of various ‘greedy’ algorithms (see [1] for details). As concerns lower bounds for the quantities (1), in the case when X = L(Ω) and Φ is an orthonormal basis in X such bounds are usually obtained by using the incompressibility property of an N -dimensional cube, which was established in [2]. With the aid of this geometric result, for a fixed m the problem can be reduced to finding a cube of dimension K ·m (where K is an absolute constant) with possibly large edge length such that all its vertices lie in F . This approach cannot be used for X = L(Ω) (see [3]). On the other hand, [3] contains rather general conditions on the set F ⊂ L(Ω) and the system Φ, which make it possible to find lower bounds for the quantities (1). One reason for taking the metric (2) lies in its connections with discrete mathematics and, in particular,