  We characterize the completeness and frame/basis property of a union of
under-sampled windowed exponentials of the form $$ {\mathcal F}(g): =\{e^{2\pi
i n x}: n\ge 0\}\cup \{g(x)e^{2\pi i nx}: n<0\} $$ for $L^2[-1/2,1/2]$ by the
spectra of the Toeplitz operators with symbol $g$. Using this characterization,
we classify all real-valued functions $g$ such that ${\mathcal F}(g)$ is
complete or forms a frame/basis. Conversely, we use the classical
Kadec-1/4-theorem in non-harmonic Fourier series to determine all $\xi$ such
that the Toeplitz operators with symbol $e^{2\pi i \xi x}$ is injective or
invertible. These results demonstrate an elegant interaction between frame
theory of windowed exponentials and Toeplitz operators. Finally, as an
application, we use our results to answer some open questions in dynamical
sampling, phase retrieval and derivative samplings on $\ell^2({\mathbb Z})$ and
Paley-Wiener spaces of bandlimited functions.
