Computing the Hilbert transform and its inverse. (English) Zbl 1226.65103
The author constructs a new method for approximating Hilbert transforms and their inverse in the complex plane. The theory is based on Riemann-Hilbert problems by using of Plemelj’s lemma. Existing approaches are rederived for computing Hilbert transforms and taken advantage that one can use the Hilbert transforms in the complex plane. First the method is demonstrated on the half line. Combining two half lines, the Hilbert transform can be computed for a more general class of functions on the real line than it is possible with existing methods.
Reviewer: Josef Kofroň (Praha)
MSC:
| 65R10 | Numerical methods for integral transforms |
| 65E05 | General theory of numerical methods in complex analysis (potential theory, etc.) |
| 30E20 | Integration, integrals of Cauchy type, integral representations of analytic functions in the complex plane |
| 44A15 | Special integral transforms (Legendre, Hilbert, etc.) |
Keywords:
Cauchy transform; Cauchy principal value integrals; Hilbert transform; Riemann-Hilbert problems; singular integral equations; quadratureReferences:
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