扇形有限生成动力框架
Sectorial Finitely Generated Dynamical Frames
- Northern Illinois University（北伊利诺伊大学）
- Université Laval（拉瓦尔大学）
- Vanderbilt University（范德堡大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文研究扇形谱约束下可逆算子生成的有限生成动力框架，通过分数轨道合成约化建立框架判据，给出密度阈值、补性质及采样条件，并利用差商预处理扩展适用范围。
AI中文摘要：
我们研究在可逆有界算子 $T$ 下由有限多个向量生成的单侧迭代框架。假设 $T$ 的谱位于 $\{\rho e^{it}:0<\rho\le1,\\ |t|\le c\}$ 中，其中 $c<\pi$，且其本质谱位于 $\{e^{it}:|t|\le c_{\mathrm e}\}$ 中，其中 $0\le c_{\mathrm e}\le c$。实幂通过主对数定义。我们的结构结果将分数轨道合成（在可逆映射和紧扰动意义下）等同于包含本质谱的弧上投影向量值指数的合成。因此，半Fredholm性质和Fredholm指标得以传递。对于一致分离的时间集 $\Lambda$，当其单侧下均匀密度满足 $D_+^-(\Lambda)>c_{\mathrm e}/\pi$ 且其对数块密度满足 $L(\Lambda)>c/\pi$ 时，该约化给出一个框架。在后一阈值处的完备性仅需一个谱位于相应角形扇区中的可逆算子以及一个完备的整数多轨道。当 $L(\Lambda)>2c/\pi$ 时，补性质等价于信道循环子空间中的有限条件，并具有实相位恢复的推论。广义差商允许相同的紧约化以及考虑重数的类似框架定理。对于单元长度 $r$ 的任意每单元单点采样，充分的原始采样条件为 $rc<\pi$ 和 $2rc_{\mathrm e}<\pi$。因此，当本质谱包含于 $\{1\}$ 时，原始采样的完整范围 $rc<\pi$ 成立；在无限制类上，尖锐的普适范围为 $rc<\pi/2$。差商预处理对所有考虑的算子恢复 $rc<\pi$，并将精确碰撞转化为对数导数数据。
英文摘要：
We study frames of unilateral iterations generated by finitely many vectors under an invertible bounded operator $T$. Suppose that the spectrum of $T$ lies in $\{ρe^{it}:0<ρ\le1,\ |t|\le c\}$, where $c<π$, and that its essential spectrum lies in $\{e^{it}:|t|\le c_{\mathrm e}\}$, where $0\le c_{\mathrm e}\le c$. Real powers are defined through the principal logarithm. Our structural result identifies fractional-orbit synthesis, up to an invertible map and a compact perturbation, with synthesis of projected vector-valued exponentials on an arc containing the essential spectrum. Semi-Fredholm properties and Fredholm index therefore transfer. For a uniformly separated temporal set $Λ$, this reduction gives a frame when its one-sided lower uniform density satisfies $D_+^-(Λ)>c_{\mathrm e}/π$ and its logarithmic block density satisfies $L(Λ)>c/π$. Completeness at the latter threshold requires only an invertible operator with spectrum in the corresponding angular sector and a complete integer multiorbit. At $L(Λ)>2c/π$, the complement property is equivalent to a finite condition in channel-cyclic subspaces, with real phase-retrieval consequences. Generalized divided differences admit the same compact reduction and the analogous frame theorem with multiplicities counted. For arbitrary one-point-per-cell sampling with cell length $r$, the sufficient raw-sampling conditions are $rc<π$ and $2rc_{\mathrm e}<π$. Thus, the full range $rc<π$ holds for raw samples when the essential spectrum is contained in $\{1\}$; over the unrestricted class, the sharp universal range is $rc<π/2$. Divided-difference preconditioning restores $rc<π$ for all operators considered and converts exact collisions into logarithmic-derivative data.
