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Matrix Cauchy and Hilbert transforms in Hermitian quaternionic Clifford analysis

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Abstract
Recently the basic setting has been established for the development of quaternionic Hermitian Clifford analysis, a theory centred around the simultaneous null solutions, called q-Hermitian monogenic functions, of four Hermitian Dirac operators in a quaternionic Clifford algebra setting. Borel–Pompeiu and Cauchy integral formulae have been established in this framework by means of a (4 × 4) circulant matrix approach. By means of the matricial quaternionic Hermitian Cauchy kernel involved in these formulae, a quaternionic Hermitian Cauchy integral may be defined. The subsequent study of the boundary limits of this Cauchy integral then leads to the definition of a quaternionic Hermitian Hilbert transform. These integral transforms are studied in this article.
Keywords
Cauchy integral, Hilbert transform, Quaternionic Hermitean Clifford analysis

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Chicago
Abreu-Blaya, Ricardo, Juan Bory-Reyes, Fred Brackx, Hennie De Schepper, and Franciscus Sommen. 2013. “Matrix Cauchy and Hilbert Transforms in Hermitian Quaternionic Clifford Analysis.” Complex Variables and Elliptic Equations 58 (8): 1057–1069.
APA
Abreu-Blaya, R., Bory-Reyes, J., Brackx, F., De Schepper, H., & Sommen, F. (2013). Matrix Cauchy and Hilbert transforms in Hermitian quaternionic Clifford analysis. COMPLEX VARIABLES AND ELLIPTIC EQUATIONS, 58(8), 1057–1069.
Vancouver
1.
Abreu-Blaya R, Bory-Reyes J, Brackx F, De Schepper H, Sommen F. Matrix Cauchy and Hilbert transforms in Hermitian quaternionic Clifford analysis. COMPLEX VARIABLES AND ELLIPTIC EQUATIONS. 2013;58(8):1057–69.
MLA
Abreu-Blaya, Ricardo, Juan Bory-Reyes, Fred Brackx, et al. “Matrix Cauchy and Hilbert Transforms in Hermitian Quaternionic Clifford Analysis.” COMPLEX VARIABLES AND ELLIPTIC EQUATIONS 58.8 (2013): 1057–1069. Print.
@article{4227806,
  abstract     = {Recently the basic setting has been established for the development of quaternionic Hermitian Clifford analysis, a theory centred around the simultaneous null solutions, called q-Hermitian monogenic functions, of four Hermitian Dirac operators in a quaternionic Clifford algebra setting. Borel--Pompeiu and Cauchy integral formulae have been established in this framework by means of a (4\,{\texttimes}\,4) circulant matrix approach. By means of the matricial quaternionic Hermitian Cauchy kernel involved in these formulae, a quaternionic Hermitian Cauchy integral may be defined. The subsequent study of the boundary limits of this Cauchy integral then leads to the definition of a quaternionic Hermitian Hilbert transform. These integral transforms are studied in this article.},
  author       = {Abreu-Blaya, Ricardo and Bory-Reyes, Juan and Brackx, Fred and De Schepper, Hennie and Sommen, Franciscus},
  issn         = {1747-6933},
  journal      = {COMPLEX VARIABLES AND ELLIPTIC EQUATIONS},
  language     = {eng},
  number       = {8},
  pages        = {1057--1069},
  title        = {Matrix Cauchy and Hilbert transforms in Hermitian quaternionic Clifford analysis},
  url          = {http://dx.doi.org/10.1080/17476933.2011.626034},
  volume       = {58},
  year         = {2013},
}

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