On the Hilbert transform of bounded bandlimited signals

In this paper we analyze the Hilbert transform and existence of the analytical signal for the space $\mathcal B_\pi^\infty$ of bandlimited signals that are bounded on the real axis. Originally, the theory was developed for signals in $L^2(\mathbb R)$ and then extended to larger signal spaces. While it is well known that the common integral representation of the Hilbert transform may diverge for some signals in $\mathcal B_\pi^\infty$ and that the Hilbert transform is not a bounded operator on $\mathcal B_\pi^\infty$, it is nevertheless possible to define the Hilbert transform for the space $\mathcal B_\pi^\infty$. We use a definition that is based on the $\mathcal H^1$–$\mathrm{BMO}(\mathbb R)$ duality. This abstract definition, which can be used for general bounded signals, gives no constructive procedure to compute the Hilbert transform. However, for the practically important special case of bounded bandlimited signals, we can provide such an explicit procedure by giving a closed-form expression for the Hilbert transform. Further, it is shown that the Hilbert transform of a signal in $\mathcal B_\pi^\infty$ is still bandlimited but not necessarily bounded. With these results we continue the work of [1,2].