Greedy approximation for biorthogonal systems in quasi-Banach spaces
Volume 560 / 2021
Abstract
The general problem addressed in this work is the development of a systematic study of the thresholding greedy algorithm for general biorthogonal systems %(also known as Markushevich bases) in quasi-Banach spaces from a functional-analytic point of view. If $(\boldsymbol{x}_n,\boldsymbol{x}_n^*)_{n=1}^\infty$ is a biorthogonal system in $\boldsymbol{X}$ then for each $x\in \boldsymbol{X}$ we have a formal expansion $\sum_{n=1}^\infty \boldsymbol{x}_n^*(x)\boldsymbol{x}_n$. The thresholding greedy algorithm (with threshold $\varepsilon \gt 0$) applied to $x$ is formally defined as $\sum_{\{n \colon |\boldsymbol{x}_n^*(x)|\geq \varepsilon\}} \boldsymbol{x}_n^*(x) \boldsymbol{x}_n$. The properties of this operator give rise to the different classes of greedy-type bases. We revisit the concepts of greedy, quasi-greedy, and almost greedy bases in this comprehensive framework and provide the (non-trivial) extensions of the corresponding characterizations of those types of bases. As a by-product of our work, new properties arise, and the relations among them are carefully discussed.
