  We further develop the concept of supergrowth [Jordan, Quantum Stud.: Math.
Found. $\textbf{7}$, 285-292 (2020)], a phenomenon complementary to
superoscillation, defined as the local amplitude growth rate of a function
being higher than its largest wavenumber. We identify the superoscillating and
supergrowing regions of a canonical oscillatory function and find the maximum
values of local growth rate and wavenumber. Next, we provide a quantitative
comparison of lengths and relevant intensities between the superoscillating and
the supergrowing regions of a canonical oscillatory function. Our analysis
shows that the supergrowing regions contain intensities that are exponentially
larger in terms of the highest local wavenumber compared to the
superoscillating regions. Finally, we prescribe methods to reconstruct a
sub-wavelength object from the imaging data using both superoscillatory and
supergrowing point spread functions. Our investigation provides an
experimentally preferable alternative to the superoscillation based
superresolution schemes and is relevant to cutting-edge research in far-field
sub-wavelength imaging.
