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 $$\Wc(f,g)=\bigset{\set{fe^{2\pi i nt}}_{n\in J} \cup \set{ge^{2\pi i nt}}_{n\in J^c}\,|\,J\subset\Z}.$$ We say that $\Wc(f,g)$ is \emph{wovenly complete}, (resp. \emph{wovenly minimal}, a \emph{woven frame}) if the weaving $\set{fe^{2\pi i nt}}_{n\in J} \cup \set{ge^{2\pi i nt}}_{n\in J^c}$ 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 $L^2(\R)$.