论文标题
框架系列的收敛
Convergence of frame series
论文作者
论文摘要
如果$ \ {x_n \} _ {n \ in \ mathbb {n}} $是Hilbert space $ h的框架，则存在一个规范的双重框架$ \ \ {\ tilde {\ tilde {x_n} {x_n} \} \} \ in \ in \ in \ mathbb { \ sum \ langle x，\ tilde {x_n} \ rangle \，x_n，$，本系列无条件收敛。但是，如果框架不是riesz的基础，则存在替代双duals $ \ {y_n \} _ {n \ in \ mathbb {n}} $和合成duals $ \ {z___n \} _ {z___n \} _ { \ rangle \，x_n，$和$ x = \ sum \ langle x，x_n \ rangle \，z_n，z_n，$对于每个$x。$，我们都会表征帧序列的框架（$ x = \ sum \ sum \ sum \ langle x，y__n \ y_n \ y_n \ rangle \ rangle \，x_n，$），$ x $ x $ x $ x $ x $ x $ x $综合二号双重双重。特别是，我们证明，如果$ \ {x_n \} _ {n \ in \ mathbb {n}} $不包含无限的许多零，则框架序列无条件地无条件地收敛于每个替代二（或合成dual），如果且仅如果$ \ \ \ \ ies a \ is a \ is a \ \ i \ \ \ in \ in \ in {接近riesz的基础。我们还证明，所有替代双重和合成二偶二重双重双重偶的过剩与它们相关的框架相同。
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.
