Extension of the Hilbert transform

The Hilbert transform is an important operator in signal processing, e.g., the definition of the “analytical signal” uses the Hilbert transform. In this paper we analyze the Hilbert transform for bounded bandlimited signals in B^∞π. Although the common integral representation of the Hilbert transform may diverge for certain signals in B^∞π, it is possible to define the Hilbert transform meaningfully for bounded signals. We employ a definition that is based on the H¹-BMO(ℝ) duality. The problem of this abstract definition is that there exists no constructive procedure to calculate the Hilbert transform. However, for the subspace of bounded bandlimited signals, we can give an explicit formula for the calculation of the Hilbert transform. Further, we show that the Hilbert transform of a bounded bandlimited signal is still bandlimited but not necessarily bounded.