Woven weighted exponentials

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 $$\text{Wc}(f,g)=\text{bigset}\{f{e}^{2\pi int}{\}}_{n\in J}\cup \{g{e}^{2\pi int}{\}}_{n\in J^c}\mathrm{\mid }J\subset \mathbb{Z}\mathrm{.}$$ We say that $\text{Wc}(f,g)$ is \emph{wovenly complete}, (resp. \emph{wovenly minimal}, a \emph{woven frame}) if the weaving $\{f{e}^{2\pi int}{\}}_{n\in J}\cup \{g{e}^{2\pi int}{\}}_{n\in J^c}$ is complete, (resp. minimal, a frame) for all $J\subseteq \mathbb{Z}\mathrm{.}$ In this paper, we study conditions that imply certain approximation properties of $\text{Wc}(f,g)$, 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 $\text{Wc}(f,g)$ is a woven frame if $f\mathrm{/}g$ is strictly positive or strictly negative over $[0,1]\mathrm{.}$ Additionally, several counterexamples are provided to show that certain seemingly correct conditions do not imply the desired approximation properties of $\text{Wc}(f,g)\mathrm{.}$ All results presented in this paper apply equivalently to systems of regular translates and Gabor systems at critical density in $L^2(\mathbb{R})$.