Characterization of the peak value behavior of the Hilbert transform of bounded bandlimited signals

The peak value of a signal is a characteristic that has to be controlled in many applications. In this paper we analyze the peak value of the Hilbert transform for the space B∞ π of bounded bandlimited signals. It is known that for this space the Hilbert transform cannot be calculated by the common principal value integral, because there are signals for which it diverges everywhere. Although the classical definition fails for B∞ π , there is a more general definition of the Hilbert transform, which is based on the abstract H1-BMO(R) duality. It was recently shown in the paper “On the Hilbert Transform of Bounded Bandlimited Signals,” Problems of Information Transmission, vol. 48, 2012 [1] that, in addition to this abstract definition, there exists an explicit formula for the calculation of the Hilbert transform. Based on this formula we study the properties of the Hilbert transform for the space B∞ π of bounded bandlimited signals. We analyze its asymptotic growth behavior, and thereby solve the peak value problem of the Hilbert transform for this space. Further, we obtain results for the growth behavior of the Hilbert transform for the subspace B∞ π,0 of bounded bandlimited signals that vanish at infinity. By studying the properties of the Hilbert transform, we continue the work of Korzhik “The extended Hilbert transformation and its application in signal theory,” Problems of Information Transmission, vol. 5, 1969 [2]. Index Terms Hilbert transform, peak value, bounded bandlimited signal, growth