Journal
MATHEMATICS OF COMPUTATION
Volume 80, Issue 275, Pages 1745-1767Publisher
AMER MATHEMATICAL SOC
DOI: 10.1090/S0025-5718-2011-02418-X
Keywords
Cauchy transform; Cauchy principal value integrals; Hilbert transform; Riemann-Hilbert problems; singular integral equations; quadrature
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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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