  In the past 50 years, quantum physicists have discovered, and experimentally
demonstrated, a phenomenon which they termed superoscillations. Aharonov and
his collaborators showed that superoscillations naturally arise when dealing
with weak values, a notion that provides a fundamentally different way to
regard measurements in quantum physics. From a mathematical point of view,
superoscillating functions are a superposition of small Fourier components with
a bounded Fourier spectrum, which result, when appropriately summed, in a shift
that can be arbitrarily large, and well outside the spectrum. Purpose of this
work is twofold: on one hand we provide a self-contained survey of the existing
literature, in order to offer a systematic mathematical approach to
superoscillations; on the other hand, we obtain some new and unexpected
results, by showing that superoscillating sequences can be seen of as solutions
to a large class of convolution equations and can therefore be treated within
the theory of Analytically Uniform spaces. In particular, we will also discuss
the persistence of the superoscillatory behavior when superoscillating
sequences are taken as initial values of the Schr\"odinger equation and other
equations.
