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Mathematics of Computation

Published by the American Mathematical Society, the Mathematics of Computation (MCOM) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.98.

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Computing the Hilbert transform and its inverse
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by Sheehan Olver PDF
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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Additional Information
  • Sheehan Olver
  • Affiliation: Numerical Analysis Group, Oxford University Mathematical Institute, 24-29 St Giles’, Oxford, England OX1 3LB
  • MR Author ID: 783322
  • ORCID: 0000-0001-6920-0826
  • Email: Sheehan.Olver@sjc.ox.ac.uk
  • Received by editor(s): November 30, 2009
  • Received by editor(s) in revised form: February 7, 2010
  • Published electronically: February 25, 2011
  • © Copyright 2011 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Math. Comp. 80 (2011), 1745-1767
  • MSC (2010): Primary 65E05, 30E20, 32A55
  • DOI: https://doi.org/10.1090/S0025-5718-2011-02418-X
  • MathSciNet review: 2785477