The natural scale of signals: Pulse duration and superoscillations

Superoscillations are oscillations at frequencies above the maximum frequency in the signal spectrum. Signals of very small bandwidth can indeed oscillate at arbitrarily high frequencies, over arbitrarily long intervals. This work addresses the matter from a different angle, emphasizing scale and discussing the following question: can an arbitrarily narrow pulse be constructed by linearly combining arbitrarily wider pulses? The connection with superoscillations and approximation theory is also discussed.