论文标题
阈值的贪婪算法与近似值的尺寸为$ f $的尺寸
The Thresholding Greedy Algorithm versus Approximations with Sizes Bounded by Certain Functions $f$
论文作者
论文摘要
令$ x $为Banach Space，$（E_N）_ {n = 1}^\ infty $为基础。对于大型集合$ \ Mathcal {F} $中的函数$ f $（在组成下关闭），我们定义并表征了$ f $ - 果岭和$ f $ - 几乎是贪婪的基础。我们将这些基础之间的关系研究为$ f $各种，表明虽然基础几乎不是贪婪，但对于某些$ f \ in \ Mathcal {f} $而言，它可能是$ f $ greedy。此外，我们证明，对于所有非身份函数$ f \ in \ mathcal {f} $，我们拥有令人惊讶的等价$$ f \ mbox { - greedy} \ \ longleftrightArrow \ f \ f \ f \ f \ f \ f \ f \ f \ f \ f \ f \ mbox { - 最大的贪婪}。$ greedy}。
Let $X$ be a Banach space and $(e_n)_{n=1}^\infty$ be a basis. For a function $f$ in a large collection $\mathcal{F}$ (closed under composition), we define and characterize $f$-greedy and $f$-almost greedy bases. We study relations among these bases as $f$ varies and show that while a basis is not almost greedy, it can be $f$-greedy for some $f\in \mathcal{F}$. Furthermore, we prove that for all non-identity function $f\in \mathcal{F}$, we have the surprising equivalence $$f\mbox{-greedy}\ \Longleftrightarrow \ f\mbox{-almost greedy}.$$ We give various examples of Banach spaces to illustrate our results.
