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Hilbert transforms in Clifford analysis

Fred Brackx (UGent) , Bram De Knock (UGent) and Hennie De Schepper (UGent)
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Abstract
The Hilbert transform on the real line has applications in many fields. In particular in one–dimensional signal processing, the Hilbert operator is used to extract global as well as instantaneous characteristics, such as frequency, amplitude and phase, from real signals. The multidimensional approach to the Hilbert transform usually is a tensorial one, considering the so-called Riesz transforms in each of the cartesian variables separately. In this paper we give an overview of generalized Hilbert transforms in Euclidean space, developed within the framework of Clifford analysis. Roughly speaking, this is a function theory of higher dimensional holomorphic functions, which is particularly suited for a treatment of multidimensional phenomena since all dimensions are encompassed at once as an intrinsic feature.
Keywords
Clifford analysis, Hilbert transform

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Chicago
Brackx, Fred, Bram De Knock, and Hennie De Schepper. 2010. “Hilbert Transforms in Clifford Analysis.” In Geometric Algebra Computing : in Engineering and Computer Science, ed. Eduardo Bayro-Corrochano and Gerik Scheuermann, 163–187. London, UK: Springer.
APA
Brackx, Fred, De Knock, B., & De Schepper, H. (2010). Hilbert transforms in Clifford analysis. In E. Bayro-Corrochano & G. Scheuermann (Eds.), Geometric algebra computing : in engineering and computer science (pp. 163–187). London, UK: Springer.
Vancouver
1.
Brackx F, De Knock B, De Schepper H. Hilbert transforms in Clifford analysis. In: Bayro-Corrochano E, Scheuermann G, editors. Geometric algebra computing : in engineering and computer science. London, UK: Springer; 2010. p. 163–87.
MLA
Brackx, Fred, Bram De Knock, and Hennie De Schepper. “Hilbert Transforms in Clifford Analysis.” Geometric Algebra Computing : in Engineering and Computer Science. Ed. Eduardo Bayro-Corrochano & Gerik Scheuermann. London, UK: Springer, 2010. 163–187. Print.
@incollection{1014170,
  abstract     = {The Hilbert transform on the real line has applications in many fields. In particular in one--dimensional signal processing, the Hilbert operator is used to extract global as well as instantaneous characteristics, such as frequency, amplitude and phase, from real signals. The multidimensional approach to the Hilbert transform usually is a tensorial one, considering the so-called Riesz transforms in each of the cartesian variables separately. In this paper we give an overview of generalized Hilbert transforms in Euclidean space, developed within the framework of Clifford analysis. Roughly speaking, this is a function theory of higher dimensional holomorphic functions, which is particularly suited for a treatment of multidimensional phenomena since all dimensions are encompassed at once as an intrinsic feature.},
  author       = {Brackx, Fred and De Knock, Bram and De Schepper, Hennie},
  booktitle    = {Geometric algebra computing : in engineering and computer science},
  editor       = {Bayro-Corrochano, Eduardo and Scheuermann, Gerik},
  isbn         = {9781849961073},
  keyword      = {Clifford analysis,Hilbert transform},
  language     = {eng},
  pages        = {163--187},
  publisher    = {Springer},
  title        = {Hilbert transforms in Clifford analysis},
  url          = {http://dx.doi.org/10.1007/978-1-84996-108-0\_9},
  year         = {2010},
}

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