2022/12/05 by Berná, Pablo M., González, David
#FOS: Mathematics #Functional Analysis (math.FA)
paper · doi:10.48550/arxiv.2212.02577
The theory of greedy-like bases started in 1999 when S. V. Konyagin and V. N. Temlyakov introduced in \citeKT the famous Thresholding Greedy Algorithm. Since this year, different greedy-like bases appeared in the literature, as for instance: quasi-greedy, almost-greedy and greedy bases. The purpose of this paper is to introduce some new characterizations of 1-greedy bases. Concretely, given a basis \mathcal B=(\mathbf xn)n∈\mathbb N in a Banach space \mathbb X, we know that \mathcal B is C-greedy with C>0 if \Vert f-\mathcal Gm(f)\Vert≤ Cσm(f) for every f∈\mathbb X and every m∈\mathbb N, where σm(f) is the best mth error in the approximation for f, that is, σm(f)=inf_y∈\mathbbX : \vert supp(y)\vert≤ m\Vert f-y\Vert. Here, we focus our attention when C=1 showing that a basis is 1-greedy if and only if \Vert f-\mathcal G1(f)\Vert=σ1(f) for every f∈\mathbb X.