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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,...
It is known that every linear method which determines the Hilbert transform from the samples of the function diverges (weakly). This paper presents strong evidence that all such methods even diverge strongly. It is shown that the common approximation method derived from the conjugate Fej'er means diverges strongly, and that all reasonable approximation methods with a finite search horizon diverge...
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...
The Hilbert transform is an important operator in signal processing, e.g., the definition of the “analytical signal” uses the Hilbert transform. In this paper we analyze the Hilbert transform for bounded bandlimited signals in B∞π. Although the common integral representation of the Hilbert transform may diverge for certain signals in B∞π, it is possible to define the Hilbert transform meaningfully...
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