arXiv 2203.070472022-03math.CAmath.FA

框架级数的收敛性

Convergence of frame series

Christopher Heil, Pu-Ting Yu

备注：: Final version

摘要：

如果$\{x_n\}_{n \in \mathbb{N}}$是一个希尔伯特空间$H,$的框架，那么存在一个规范的对偶框架$\{\tilde{x_n}\}_{n \in \mathbb{N}}$，使得对于每个$x \in H$，我们都有$x = \sum \langle x, \tilde{x_n} \rangle \, x_n,$，这个级数无条件收敛。然而，如果该框架不是Riesz基，则存在替代的$\{y_n\}_{n \in \mathbb{N}}$和合成-伪$\{z_n\}_{n \in \mathbb{N}}$，使得对于每个$x.$，$x = \sum \langle x, y_n \rangle \, x_n,$和$x = \sum \langle x, x_n \rangle \, z_n,$我们表征了对于每个$x$，对于每个替代对偶，并且对于合成-伪对偶，框架序列（$x = \sum \langle x, y_n \rangle \, x_n,$）无条件收敛的框架。特别是，我们证明了，如果$\{x_n\}_{n \in \mathbb{N}}$不包含无限多个零，那么框架系列无条件收敛的每一个替代的对偶（或合成伪对偶）当且仅当$\{x_n\}_{n \in \mathbb{N}}$是一个近Riesz基础。我们还证明了所有的替代框架和合成伪框架都有与它们相关联的框架相同的过剩。

英文摘要：

If $\{x_n\}_{n \in \mathbb{N}}$ is a frame for a Hilbert space $H,$ then there exists a canonical dual frame $\{\tilde{x_n}\}_{n \in \mathbb{N}}$ such that for every $x \in H$ we have $x = \sum \langle x, \tilde{x_n} \rangle \, x_n,$ with unconditional convergence of this series. However, if the frame is not a Riesz basis, then there exist alternative duals $\{y_n\}_{n \in \mathbb{N}}$ and synthesis-pseudo duals $\{z_n\}_{n \in \mathbb{N}}$ such that $x = \sum \langle x, y_n \rangle \, x_n,$ and $x = \sum \langle x, x_n \rangle \, z_n,$ for every $x.$ We characterize the frames for which the frame series ($x = \sum \langle x, y_n \rangle \, x_n,$) converges unconditionally for every $x$ for every alternative dual, and similarly for synthesis-pseudo duals. In particular, we prove that if $\{x_n\}_{n \in \mathbb{N}}$ does not contain infinitely many zeros then the frame series converge unconditionally for every alternative dual (or synthesis-pseudo dual) if and only if $\{x_n\}_{n \in \mathbb{N}}$ is a near-Riesz basis. We also prove that all alternative duals and synthesis-pseudo duals have the same excess as their associated frame.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

摘要：

英文摘要：

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