  Superoscillations, i.e., the phenomenon that a bandlimited function can
temporary oscillate faster than its highest Fourier component, are being much
discussed for their potential for `superresolution' beyond the diffraction
limit. Here, we consider systems that are driven with a time dependence that is
off-resonance for the system, in the Fourier sense. We show that
superoscillating sources can temporarily induce resonance during the period
when the source is behaving superoscillatory. This observation poses the
question as to how the system `undoes' the `false resonance' after the full
source has acted and its band limitation is apparent. We discuss several
examples of systems which might be capable of distilling the temporary
excitation through some non-harmonic effects, such as dissipation or dispersion
at high frequencies, opening up the possibility of low frequency detection of
`fast' microphysics through superoscillations. We conclude that, either
superoscillations really can beat the bandlimit and achieve superresolution
(`kinematic superresolution') or the superoscillating high frequency is
absorbed and we gain dynamical access to the physics of high frequency
processes with low frequency signals (`dynamical superresolution').
