Spectra of Bounded Functions

We are concerned here with waveforms x(t), -∞ < t < ∞, which satisfy an amplitude-constraint, |x(t)| ≤ A < ∞, and their spectra. We pose two open problems. The first is the maximization of the energy of a filtered version of an amplitude-constrained pulse with finite support. The second is the question of how close the power spectral density of a stationary amplitude-constrained random process can be to a flat band-limited spectrum. These questions appear to be difficult, but answers to them will shed light on certain aspects of storage in magnetic media (disks, tapes, etc. which are inherently amplitude limited) and on communication over microwave radio links.