Seminar on Banach and Metric Space Geometry

Date: November 10, 2022

Time: 10:00AM - 11:00AM

Location: BLOC 302

Speaker: Hung Viet Chu, University of Illinois at Urbana-Champaign

Title: New Greedy-type Bases from Comparing Greedy Sums against Other Approximants of Different Sizes

Abstract: One goal of approximation theory is to approximate a vector $x$ using finite linear combinations of vectors, called approximants, in a given basis. For higher efficiency, the approximants can be made adaptive to the vector $x$ as in the thresholding greedy algorithm (TGA) introduced by Konyagin and Temlyakov in 1999. The optimality of the TGA is captured by the notion of greedy and almost greedy bases. A basis $(e_n)_{n=1}^\infty$ of a Banach space $X$ (over a field $\mathbb{F}$) is said to be greedy if there exists a constant $\mathbf C\geqslant 1$ such that $$\|x-G_m(x)\|\ \leqslant\ \mathbf C\inf_{\substack{|A|\leqslant m\\(a_n)_{n\in A}\subset \mathbb{F}}}\left\|x-\sum_{n\in A}a_ne_n\right\|.$$ Here, $G_m(x)$ is an approximant of size $m$ given by the TGA. The definition of almost greedy bases replaces the arbitrary linear combinations on the right by projections. Extending classical results, we define ($f$, greedy) bases to satisfy the condition: there exists a constant $\mathbf C\geqslant 1$ such that $$\|x-G_m(x)\|\ \leqslant\ \mathbf C\inf_{\substack{|A|\leqslant f(m)\\(a_n)_{n\in A}\subset \mathbb{F}}}\left\|x-\sum_{n\in A}a_ne_n\right\|,$$ where $f$ belongs to $\mathcal{F}$, a collection that contains functions like $f(x) = cx^{\gamma}$ for $c, \gamma\in [0,1]$. The definition of ($f$, almost greedy) is modified accordingly. We characterize these bases and establish the surprising equivalence: if $f$ is a non-identity function in $\mathcal{F}$, then a basis is ($f$, greedy) if and only if it is ($f$, almost greedy). We show that ($f$, greedy) bases form a much wider class as there exist examples of classical bases that are not almost greedy but is ($f$, greedy) for some $f\in\mathcal{F}$. Besides greedy and almost greedy bases, we also have partially greedy bases, which compare $G_m(x)$ against partially summations. Inspired by a theorem due to Dilworth, Kalton, Kutzarova, and Temlyakov, we show that for a fixed $\lambda > 1$, replacing $G_m(x)$ by $G_{\lceil\lambda m\rceil}(x)$ in the defini