Complete characterization of bandlimited signals with bounded Hilbert transform

For the space of bounded bandlimited signals a definition of the Hilbert transform by the usual Hilbert transform integral is not possible, because the integral diverges for certain bounded bandlimited signals. There are other ways to define the Hilbert transform meaningfully. Recently, it was shown that, for bounded bandlimited signals, a simple formula can be used to calculate the Hilbert transform. However, the Hilbert transform of a bounded bandlimited signal is not necessarily bounded again. In this paper, we completely characterize the bounded bandlimited signals that have a bounded Hilbert transform by giving a necessary and sufficient condition for the boundedness. Further, we use this condition to prove that there exist bounded bandlimited signals that even vanish at infinity, the Hilbert transform of which is unbounded.