The results are separate, and apparently paradoxical, and have implications for physics.  First, when two exponentials compete, their interference can be dominated by the contribution with smaller exponent.  Second, repeated differentiation of almost all functions in a wide class generates trigonometric oscillations (‘almost all functions tend to cosx’).  Third, it is possible to find band-limited functions that oscillate arbitrarily faster than their fastest Fourier component (‘superoscillations’).