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 that 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”).