[Paper Review] Characterization of $1$-almost greedy bases

Fernando Albiac, José L. Ansorena|arXiv (Cornell University)|Jun 10, 2015

Advanced Banach Space Theory4 references3 citations

TL;DR

This paper provides a complete characterization of 1-almost greedy bases in Banach spaces by introducing and proving that such bases are precisely those possessing Property (A). The result closes a series of isometric characterizations initiated by Albiac and Wojtaszczyk (2006) and continued by Albiac, Ansorena, and others, showing that a basis is 1-greedy if and only if it is both 1-almost greedy and 1-quasi-greedy.

ABSTRACT

This article closes the cycle of characterizations of greedy-like bases in the isometric case initiated in [F. Albiac and P. Wojtaszczyk, Characterization of $1$-greedy bases, J. Approx. Theory 138 (2006)] with the characterization of $1$-greedy bases and continued in [F. Albiac and J. L. Ansorena, Characterization of $1$-quasi-greedy bases, arXiv:1504.04368v1 [math.FA] (2015)] 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 (semi-normalized) 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.

Motivation & Objective

• To close the cycle of isometric characterizations of greedy-like bases by fully characterizing 1-almost greedy bases.
• To establish that a basis is 1-almost greedy if and only if it satisfies Property (A).
• To clarify the relationship between 1-greedy, 1-quasi-greedy, and 1-almost greedy bases.
• To provide a tight characterization of almost greedy bases using Property (A) as a unifying criterion.
• To extend the validity of the results beyond Schauder bases to general semi-normalized bounded biorthogonal systems.

Proposed method

• Introduces and defines Property (A) for a basis in a Banach space, a condition involving the norm of unions of basis elements with signs.
• Uses the natural greedy ordering for each element x ∈ X to define the m-th greedy approximation G_m(x).
• Applies the theory of democratic and unconditional bases, particularly focusing on the 1-case where constants are optimal.
• Employs duality and renorming techniques to analyze the behavior of the greedy algorithm and its approximation error.
• Leverages prior characterizations of 1-greedy and 1-quasi-greedy bases to derive the equivalence between 1-almost greedy bases and Property (A).
• Validates the results in the broader context of semi-normalized bounded biorthogonal systems, not just Schauder bases.

Experimental results

Research questions

• RQ1 What is the complete isometric characterization of 1-almost greedy bases in Banach spaces?
• RQ2 How does Property (A) relate to the almost greedy property when the almost greedy constant is 1?
• RQ3 Can the class of 1-greedy bases be fully characterized as the intersection of 1-almost greedy and 1-quasi-greedy bases?
• RQ4 Is Property (A) sufficient and necessary for a basis to be 1-almost greedy?
• RQ5 Does the characterization extend to general bounded biorthogonal systems beyond Schauder bases?

Key findings

• A basis in a Banach space is 1-almost greedy if and only if it has Property (A).
• The 1-greedy constant is 1 if and only if the basis is both 1-almost greedy and 1-quasi-greedy.
• Property (A) is a necessary and sufficient condition for a basis to be 1-almost greedy, providing a tight characterization.
• The characterization of 1-almost greedy bases completes the isometric classification program initiated by Albiac and Wojtaszczyk (2006) and extended by Albiac, Ansorena, and others.
• The results are valid not only for Schauder bases but also for general semi-normalized bounded biorthogonal systems.
• Failure of Property (A) prevents a basis from being 1-almost greedy, even if it is 1-lattice unconditional and 1-superdemocratic, as shown by counterexamples in prior literature.