The best m-term approximation and greedy algorithms

Abstract

Two theorems on nonlinear $m$‐term approximation in $L_p ,\;1 < p < \infty$, are proved in this paper. The first one (theorem 2.1) says that if a basis ${\Psi }:{ = }\left\{ {\psi _I } \right\}_I$ is $L_p$‐equivalent to the Haar basis then a near best $m$>‐term approximation to any $f \in L_p$ can be realized by the following simple greedy type algorithm. Take the expansion $f = \sum\nolimits_I {c_I \psi _I }$ and form a sum of $m$ terms with the largest $\left\| {c_I \psi _I } \right\|_p$ out of this expansion.

The second one (theorem 3.3) states that nonlinear $m$‐term approximations with regard to two dictionaries: the Haar basis and the set of all characteristic functions of intervals are equivalent in a very strong sense.