Peak value blowup of approximations of the Hilbert transform of signals with finite energy

Abstract

Although the Hilbert transform plays an important role in many different applications, it is usually impossible to calculate it exactly in closed form. Therefore approximation methods are applied to determine numerically the Hilbert transform. The present paper studies a general class of approximation methods on signal spaces of finite Dirichlet energy. This class is characterized by two natural axioms, and it is shown that convergent methods only exist if the signal energy is sufficiently concentrated in the low frequency components of the signal. Otherwise all approximation methods which are based on discrete samples of the signal show a peak value blowup behavior, i.e. the maximum value of the approximation gets arbitrarily large as the approximation degree increases for some signals in the space.