On the convergence conditions of a weak greedy algorithm

This report is devoted to the convergence of a weak greedy algorithm, which is used to find an m-term approximation of an arbitrary element of a Hilbert space by elements of a normalized dictionary. V. N. Temlyakov and E. D. Livshits obtained convergence conditions for this algorithm depending on the behavior of the elements of the weakening sequence. We present new necessary and sufficient conditions for the convergence of a weak greedy algorithm, as well as a criterion for its convergence in the case of a quasi-monotone weakening sequence.