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On the interplay between the Hilbert transform and conjugate harmonic functions

✍ Scribed by Fred Brackx; Bram De Knock; Hennie De Schepper; David Eelbode

Publisher: John Wiley and Sons
Year: 2006
Tongue: English
Weight: 148 KB
Volume: 29
Category: Article
ISSN: 0170-4214
DOI: 10.1002/mma.735

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✦ Synopsis

As is well-known, there is a close and well-deÿned connection between the notions of Hilbert transform and of conjugate harmonic functions in the context of the complex plane. This holds e.g. in the case of the Hilbert transform on the real line, which is linked to conjugate harmonicity in the upper (or lower) half plane. It also can be rephrased when dealing with the Hilbert transform on the boundary of a simply connected domain related to conjugate harmonics in its interior (or exterior). In this paper, we extend these principles to higher dimensional space, more speciÿcally, in a Cli ord analysis setting. We will show that the intimate relation between both concepts remains, however giving rise to a range of possibilities for the deÿnition of either new Hilbert-like transforms, or speciÿc notions of conjugate harmonicity. Copyright ? 2006 John Wiley & Sons, Ltd.

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