  We propose a spectral-averaging procedure that enables computation of
bandwidth-integrated local density of states (LDOS) from a single scattering
calculation, and exploit it to investigate the minimum extinction achievable
from dipolar sources over finite bandwidths in structured media.
Structure-agnostic extinction bounds are derived, providing analytical insights
into scaling laws and fundamental design tradeoffs with implications to
bandwidth and material selection. We find that perfect LDOS suppression over a
finite bandwidth $\Delta\omega$ is impossible. Inspired by limits which predict
nontrivial $\sqrt{\Delta\omega}$ scaling in systems with material dissipation,
we show that pseudogap edge states of quasi-1d bullseye gratings can -- by
simultaneously minimizing material absorption and radiation -- yield
arbitrarily close to perfect LDOS suppression in the limit of vanishing
bandwidth.
