Sharp sufficient condition for the convergence of greedy expansions with errors in coefficient computation

Abstract

Abstract Generalized approximate weak greedy algorithms (gAWGAs) were introduced by Galatenko and Livshits as a generalization of approximate weak greedy algorithms, which, in turn, generalize weak greedy algorithm and thus pure greedy algorithm. We consider a narrower case of gAWGA in which only a sequence of absolute errors { ξ n } n = 1 ∞ {\left\{{\xi }_{n}\right\}}_{n=1}^{\infty } is nonzero. In this case sufficient condition for a convergence of a gAWGA expansion to an expanded element obtained by Galatenko and Livshits can be written as ∑ n = 1 ∞ ξ n 2 < ∞ {\sum }_{n=1}^{\infty }{\xi }_{n}^{2}\lt \infty . In the present article, we relax this condition and show that the convergence is guaranteed for ξ n = o 1 n {\xi }_{n}=o\left(\frac{1}{\sqrt{n}}\right) . This result is sharp because the convergence may fail to hold for ξ n ≍ 1 n {\xi }_{n}\hspace{0.33em}\asymp \hspace{0.33em}\frac{1}{\sqrt{n}} .