织状加权指数
Woven weighted exponentials
中文总结 AI 辅助
本文研究织状加权指数系统 $\text{Wc}(f,g)$ 的近似性质，给出其织状完备性的完整刻画，证明 $f/g$ 严格正负时其为织状框架，还给出反例并推广至正则平移与临界密度 Gabor 系统。
中文摘要 AI 辅助
设 $f$ 和 $g$ 是 $L^2([0,1])$ 中的非零函数，与 $f$ 和 $g$ 相关的织状加权指数系统（记为 $\boldsymbol{\text{Wc}(f,g)}$）定义为：$\boldsymbol{\text{Wc}(f,g)=\big\{\big\{fe^{2\pi i nt}\big\}_{n\in J} \cup \big\{ge^{2\pi i nt}\big\}_{n\in J^c}\mid J\subset\Z\big\}}$。若对所有 $J\subseteq\Z$，织集 $\big\{fe^{2\pi i nt}\big\}_{n\in J} \cup \big\{ge^{2\pi i nt}\big\}_{n\in J^c}$ 在 $L^2([0,1])$ 中是完备的（或极小的、是一个框架），则称 $\text{Wc}(f,g)$ 是织状完备的（或织状极小的、是一个织状框架）。本文研究能使 $\text{Wc}(f,g)$ 具备完备性、极小性和框架性等近似性质的条件：首先给出了织状加权指数系统是织状完备的完整刻画；还证明了当 $f/g$ 在 $[0,1]$ 上严格为正或严格为负时，$\text{Wc}(f,g)$ 是一个织状框架；此外，给出了若干反例，表明某些看似正确的条件并不能使 $\text{Wc}(f,g)$ 具备所需的近似性质。本文所有结果等价适用于 $L^2(\boldsymbol{\text{R}})$ 中的正则平移系统和临界密度下的 Gabor 系统。
英文摘要
Let $f$ and $g$ be nonzero functions in $L^2([0,1])$. The \emph{woven weighted exponential system} (associated with $f$ and $g$) is defined by $$\Wc(f,g)=\bigset{\set{fe^{2πi nt}}_{n\in J} \cup \set{ge^{2πi nt}}_{n\in J^c}\,|\,J\subset\Z}.$$ We say that $\Wc(f,g)$ is \emph{wovenly complete}, (resp. \emph{wovenly minimal}, a \emph{woven frame}) if the weaving $\set{fe^{2πi nt}}_{n\in J} \cup \set{ge^{2πi nt}}_{n\in J^c}$ is complete, (resp. minimal, a frame) for all $J\subseteq \Z.$ In this paper, we study conditions that imply certain approximation properties of $\Wc(f,g)$, such as completeness, minimality and the frame property. We first provide a complete characterization of the woven weighted exponential systems that are wovenly complete. We also show that $\Wc(f,g)$ is a woven frame if $f/g$ is strictly positive or strictly negative over $[0,1].$ Additionally, several counterexamples are provided to show that certain seemingly correct conditions do not imply the desired approximation properties of $\Wc(f,g).$ All results presented in this paper apply equivalently to systems of regular translates and Gabor systems at critical density in $L^2(\R)$.
