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

Valiullin Artur R., Valiullin Albert R., Solodov Alexei P.

Demonstratio Mathematica

2022

Vol. 55, nr 1

254--264

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 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=1ξ2n<∞ . In the present article, we relax this condition and show that the convergence is guaranteed for ξn=o(1√n) . This result is sharp because the convergence may fail to hold for ξn≍1√n .