Characterization of 1-almost greedy bases

Abstract

Abstract. This article closes the cycle of characterizations of greedy-like bases in the "isometric" case initiated in [1] with the characterization of 1-greedy bases and continued in [2] with the characterization of 1-quasi-greedy bases. Here we settle the problem of providing a characterization of 1-almost greedy bases in Banach spaces. We show that a basis in a Banach space is almost greedy with almost greedy constant equal to 1 if and only if it has Property (A). This fact permits now to state that a basis is 1-greedy if and only if it is 1-almost greedy and 1-quasi-greedy. As a by-product of our work we also provide a tight characterization of almost greedy 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.…”

The best m-term approximation with respect to polynomials with constant coefficients

“…The concepts of suppression unconditional and symmetric for largest coefficients bases can be found in [2,3,4,6,13]. We recall here that a basis is K s -suppression unconditional if the projection operator is uniformly bounded, that is to say…”

The weighted property (A) and the greedy algorithm

“…We denoted by C q the least constant that satisfies (2), it is called the quasi-greedy constant and we say that B is C q -quasi-greedy.…”