Computing the Hilbert transform and its inverse

Olver, Sheehan

doi:10.1090/S0025-5718-2011-02418-X

Computing the Hilbert transform and its inverse
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Math. Comp. 80 (2011), 1745-1767 Request permission

Abstract:

We construct a new method for approximating Hilbert transforms and their inverse throughout the complex plane. Both problems can be formulated as Riemann–Hilbert problems via Plemelj’s lemma. Using this framework, we rederive existing approaches for computing Hilbert transforms over the real line and unit interval, with the added benefit that we can compute the Hilbert transform in the complex plane. We then demonstrate the power of this approach by generalizing to the half line. Combining two half lines, we can compute the Hilbert transform of a more general class of functions on the real line than is possible with existing methods.

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