  An accurate treatment of electronic spectra in large systems with a technique
such as time dependent density functional theory (TDDFT) is computationally
challenging. Due to the Nyquist sampling theorem, direct real time simulations
must be prohibitively long in order to recover a suitably sharp resolution in
frequency space. Super-resolution techniques such as compressed sensing and
MUSIC assume only a small number of excitations contribute to the spectrum,
which fails in large molecular systems where the number of excitations is
typically very large. We present a new approach that combines exact short time
dynamics with approximate frequency space methods to capture large narrow
features embedded in a dense manifold of smaller nearby peaks. We show that our
approach can accurately capture narrow features and broad quasi-continuum of
states at the same time - even when the features overlap in frequency. Our
approach is able reduce the required simulation time by a factor of 20-40 with
respect to standard Fourier analysis and shows promise for the accurate
whole-spectrum prediction of large molecules and materials.
