Weak Greedy Algorithms for Nonlinear Approximation with Quasi-Greedy Bases

摘要

We study weak greedy approximation with respect to quasi-greedy bases. For a Hilbert space, we prove that the error of the m-th weak greedy approximation is bounded by the error of best m-term approximation multiplied by an extra factor of order ln m. However we show weak greedy algorithm realizes the best expansional m-term approximation for individual element and best m-term approximation for some sparse classes. Furthermore we establish sharp Lebesgue-type inequality for expansional m-term approximation with quasi-greedy bases in Banach spaces.