A matrix Hilbert transform in Hermitean Clifford analysis

• Published in: Journal of mathematical analysis and applications Vol. 344; no. 2; pp. 1068 - 1078
• Main Authors: Brackx, F., De Knock, B., De Schepper, H.
• Format: Journal Article
• Language: English
• Published: San Diego, CA Elsevier Inc 15-08-2008; Elsevier
• Subjects: Algebra; Cauchy integral; Exact sciences and technology; Hermitean Clifford analysis; Hilbert transform; Linear and multilinear algebra, matrix theory; Mathematical analysis; Mathematics; Sciences and techniques of general use; Several complex variables and analytic spaces; Hilbert transform; Cauchy integral; Hermitean Clifford analysis; Harmonic analysis; Cauchy integral;Hilbert transform;Hermitean Clifford analysis; Mathematical analysis; Application; Matrix function
• ISSN: 0022-247X; 1096-0813
• DOI: 10.1016/j.jmaa.2008.03.043

Orthogonal Clifford analysis is a higher dimensional function theory offering both a generalization of complex analysis in the plane and a refinement of classical harmonic analysis. During the last years, Hermitean Clifford analysis has emerged as a new and successful branch of it, offering yet a refinement of the orthogonal case. Recently in [F. Brackx, B. De Knock, H. De Schepper, D. Peña Peña, F. Sommen, submitted for publication], a Hermitean Cauchy integral was constructed in the framework of circulant
(
2
×
2
)
matrix functions. In the present paper, a new Hermitean Hilbert transform is introduced, arising naturally as part of the non-tangential boundary limits of that Hermitean Cauchy integral. The resulting matrix operator is shown to satisfy properly adapted analogues of the characteristic properties of the Hilbert transform in classical analysis and orthogonal Clifford analysis.