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