  We experimentally investigate the sub-Fourier behavior of a $\delta$-kicked
rotor resonance by performing a measurement of the fidelity or overlap of a
Bose-Einstein condensate (BEC) exposed to a periodically pulsed standing wave.
The temporal width of the fidelity resonance peak centered at the Talbot time
and zero initial momentum exhibits an inverse cube pulse number ($1/N^{3}$)
dependent scaling compared to a $1/N^{2}$ dependence for the mean energy width
at the same resonance. A theoretical analysis shows that for an accelerating
potential the width of the resonance in acceleration space depends on
$1/N^{3}$, a property which we also verify experimentally. Such a sub-Fourier
effect could be useful for high precision gravity measurements.
