Characterization of the range of the Hilbert transform for bounded bandlimited signals and applications

Recently, a new constructive formula for the calculation of the Hilbert transform of bounded bandlimited signals was found. In this paper we use that formula to analyze the properties of the Hilbert transform. We further present a Fefferman-Stein-type decomposition theorem for bandlimited signals in BMO(R), i.e., bandlimited signals of bounded mean oscillation. Based on this decomposition we characterize the range of the Hilbert transform and derive properties of general bandlimited signals in BMO(R). We show the boundedness of bandpass signals in BMO(R) and the boundedness of the derivative of bandlimited signals in BMO(R). We further find the maximum growth of the Hilbert transform of bounded bandlimited signals.