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 Bπ∞ of bandlimited signals that are bounded on the real axis. Originally, the theory was developed for signals in L2(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 Bπ∞ and that the Hilbert transform is not a bounded operator on Bπ∞, it is nevertheless possible to define the Hilbert transform for the space Bπ∞. We use a definition that is based on the H1-BMO(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 Bπ∞ is still bandlimited but not necessarily bounded. With these results we continue the work of [1, 2].
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