Efficiency of weak greedy algorithms for m-term approximations

Abstract

We investigate the efficiency of weak greedy algorithms for $m$-term expansional approximation with respect to quasi-greedy bases in general Banach spaces. We estimate the corresponding Lebesgue constants for the weak thresholding greedy algorithm (WTGA) and weak Chebyshev thresholding greedy algorithm. Then we discuss the greedy approximation on some function classes. For some sparse classes induced by uniformly bounded quasi-greedy bases of $L_p$, $1＜ p＜\infty$, we show that the WTGA realizes the order of the best $m$-term approximation. Finally, we compare the efficiency of the weak Chebyshev greedy algorithm (WCGA) with the thresholding greedy algorithm (TGA) when applying them to quasi-greedy bases in $L_p,\,1\leq p＜\infty$, by establishing the corresponding Lebesgue-type inequalities. It seems that when $p＞2$ the WCGA is better than the TGA.