  Autoionizing resonances that arise from the interaction of a bound
single-excitation with the continuum can be accurately captured with the
presently used approximations in time-dependent density functional theory
(TDDFT), but those arising from a bound double excitation cannot. In the former
case, we explain how an adiabatic kernel, which has no frequency-dependence,
can yet generate the strongly frequency-dependent resonant structures in the
interacting response function, not present in the Kohn-Sham response function.
In the case of the bound double-excitation, we explain that a strongly
frequency-dependent kernel is needed, and derive one for the vicinity of a
resonance of the latter type, as an {\it a posteriori} correction to the usual
adiabatic approximations in TDDFT. Our approximation becomes exact for an
isolated resonance in the limit of weak interaction, where one discrete state
interacts with one continuum. We derive a "Fano TDDFT kernel" that reproduces
the Fano lineshape within the TDDFT formalism, and also a dressed kernel, that
operates on top of an adiabatic approximation. We illustrate our results on a
simple model system.
