{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,8]],"date-time":"2026-05-08T21:49:03Z","timestamp":1778276943816,"version":"3.51.4"},"reference-count":28,"publisher":"American Mathematical Society (AMS)","issue":"275","license":[{"start":{"date-parts":[[2012,2,25]],"date-time":"2012-02-25T00:00:00Z","timestamp":1330128000000},"content-version":"am","delay-in-days":365,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>We construct a new method for approximating Hilbert transforms and their inverse throughout the complex plane. Both problems can be formulated as Riemann\u2013Hilbert problems via Plemelj\u2019s 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.<\/p>","DOI":"10.1090\/s0025-5718-2011-02418-x","type":"journal-article","created":{"date-parts":[[2011,2,25]],"date-time":"2011-02-25T08:38:43Z","timestamp":1298623123000},"page":"1745-1767","source":"Crossref","is-referenced-by-count":42,"title":["Computing the Hilbert transform and its inverse"],"prefix":"10.1090","volume":"80","author":[{"given":"Sheehan","family":"Olver","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"14","published-online":{"date-parts":[[2011,2,25]]},"reference":[{"key":"1","series-title":"Cambridge Texts in Applied Mathematics","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511791246","volume-title":"Complex variables: introduction and applications","author":"Ablowitz, Mark J.","year":"2003","ISBN":"https:\/\/id.crossref.org\/isbn\/0521534291","edition":"2"},{"key":"2","series-title":"National Bureau of Standards Applied Mathematics Series, No. 55","volume-title":"Handbook of mathematical functions with formulas, graphs, and mathematical tables","author":"Abramowitz, Milton","year":"1964"},{"key":"3","doi-asserted-by":"crossref","unstructured":"S. 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