Lebesgue constants for the weak greedy algorithm

Equivalence between almost-greedy and semi-greedy bases

“…As discussed in [8] (see also [4]), one can define the Thresholding Greedy Algorithm and the Thresholding Chebyshev Greedy Algorithm in the context of Markushevich bases, that is, {e i , e * i } is a semi-normalized biorthogonal system, X = span{e i : i ∈ N} X and X * = span{e * i : i ∈ N} w * . In section a) of Theorem 1.10, it is enough to work with Markushevich bases instead of Schauder bases.…”

Lebesgue constants for Chebyshev thresholding greedy algorithms

“…Examples of quasi-greedy bases can be found in the literature [4][5][6][7][8][9]. Of course, bases need not to be quasi-greedy, there exists a non-quasi-greedy basis, for these types of bases, TGA may fail to converge for certain vector x ∈ X.…”

“…In [6] and [12], the authors got the exact orders for L ch n ( ) and L ch n ( ) with respect to quasi-greedy bases. To state their results, we recall some notions.…”

Lebesgue inequalities for Chebyshev thresholding greedy algorithms

“…The second result is a slight generalization of [9,Theorem 4.1], and gives a correct version of [18,Theorem 3.5]. Theorem 1.2.…”

“…These were introduced by Dilworth, Kalton and Kutzarova, see [7, §3], as an enhancement of the TGA. Here, we use the weak version considered in [9]. Namely, for fixed t ∈ (0, 1] we say that CG t m : X → X is a Chebyshev t-greedy operator of order m if for every x ∈ X the set A = supp CG t m (x) ∈ G(x, m,t) and moreover…”