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

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 {ξn}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∞ξn2<∞{\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 ξn=o1n{\xi }_{n}=o\left(\frac{1}{\sqrt{n}}\right). This result is sharp because the convergence may fail to hold for ξn≍1n{\xi }_{n}\hspace{0.33em}\asymp \hspace{0.33em}\frac{1}{\sqrt{n}}.

Keywords

• greedy expansion
• prescribed coefficients
• hilbert space
• greedy approximation
• convergence
• 41-xx
• 41a58