Undersampled windowed exponentials and their applications

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.