Characterization of 1-greedy bases

Abstract

A basis for a Banach space X is greedy if and only if the greedy algorithm provides, up to a constant C depending only on X, the best m-term approximation for each element of the space. It is known that the Haar (or good wavelet) basis is a greedy basis in L p (0, 1) for 1 < p < ∞ [V.N. Temlyakov, The best m-term approximation and greedy algorithms, Adv. in Comp. Math. 8 (1998) 249-265]. In this particular example, unfortunately, the constant of greediness C = C(p) is strictly bigger than 1 unless p = 2. Our goal is to investigate 1-greedy bases, i.e., bases for which the greedy algorithm provides the best m-term approximation. We find a characterization of 1-greediness, study how 1-greedy bases relate to symmetric bases, and show that 1-greediness does not imply 1-symmetry, answering thus two questions raised in [P.

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.…”

“…Moreover, it is shown in [5] that if the quasi-greedy constant K = 1, then is an unconditional basis. So compared to O(K 3 ) in the upper bounds of Theorem 1.3, our results improve the implicit constants for all conditional quasi-greedy bases since in this case K > 1.…”

Characterization of 1-almost greedy bases

“…It is not hard to prove that if B is C-symmetric for largest coefficients and y is a greedy permutation of x then y ≤ C 2 x . In particular, if B has Property (A) and y is a greedy permutation of x then y = x (which is the way Property (A) was originally defined in [1]). …”

“…This study was initiated in [1], where the authors obtained the following characterization of 1-greedy bases.…”

Greedy Type Bases in Banach Spaces

“…However this is not the characterization of bases with greedy constant 1 (see [40]). The problem of isometric characterization has been solved recently in [2]. To state the result we have to introduce the so called Property (A).…”