  Superoscillating functions, i.e., functions that locally oscillate at a rate
faster than their highest Fourier component, are of interest for applications
from fundamental physics to engineering. Here, we develop a new method which
allows one to construct superoscillations of arbitrarily high frequency and
arbitrarily long duration in a computationally efficient way. We also present a
method for constructing non-singular Schr\"odinger potentials whose ground
state is a superoscillating wave function.
