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Since, for certain bounded signals, the common integral definition of the Hilbert transform may diverge, it was long thought that the Hilbert transform does not exist for general bounded signals. However, using a definition that is based on the H1-BMO(R) duality, it is possible to define the Hilbert transform meaningfully for the space of bounded signals. Unfortunately, this abstract definition gives no constructive procedure for the calculation of the Hilbert transform. However, if the signals are additionally bandlimited, i.e., if we consider signals in Bπ∞, it was recently shown that an explicit formula for the calculation of the Hilbert transform does exist. Based on this result, we analyze the asymptotic growth behavior of the Hilbert transform of signals in Bπ∞ and solve the peak value problem of the Hilbert transform. It is shown that the order of growth of Hilbert transform of signals in Bπ∞ is at most logarithmic.