Surprisingly, differentiable functions are able to oscillate arbitrarily faster than their
highest Fourier component would suggest. The phenomenon is called superoscillation.
Recently, a practical method for calculating superoscillatory functions was
presented and it was shown that superoscillatory quantum mechanical wave functions
should exhibit a number of counter-intuitive physical effects. Following up on
this work, we here present more general methods which allow the calculation of
superoscillatory wave functions with custom-designed physical properties. We give
concrete examples and we prove results about the limits to superoscillatory behavior.
We also give a simple and intuitive new explanation for the exponential computational
cost of superoscillations