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 trans-form. In this paper we analyze the Hilbert transform for bounded bandlimited signals in B∞π. Although the common integral repre-sentation 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 trans-form. 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. Index Terms — Hilbert transform, bounded bandlimited signal, bounded mean oscillation, Hardy spac