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Mathematics of Computation

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Computing the Hilbert transform and its inverse


Author: Sheehan Olver
Journal: Math. Comp. 80 (2011), 1745-1767
MSC (2010): Primary 65E05, 30E20, 32A55
DOI: https://doi.org/10.1090/S0025-5718-2011-02418-X
Published electronically: February 25, 2011
MathSciNet review: 2785477
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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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Additional Information

Sheehan Olver
Affiliation: Numerical Analysis Group, Oxford University Mathematical Institute, 24-29 St Giles’, Oxford, England OX1 3LB
Email: Sheehan.Olver@sjc.ox.ac.uk

DOI: https://doi.org/10.1090/S0025-5718-2011-02418-X
Keywords: Cauchy transform, Cauchy principal value integrals, Hilbert transform, Riemann–Hilbert problems, singular integral equations, quadrature
Received by editor(s): November 30, 2009
Received by editor(s) in revised form: February 7, 2010
Published electronically: February 25, 2011
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.