Complex Analysis and Operator Theory

, Volume 6, Issue 5, pp 987–1010 | Cite as

Phase Derivative of Monogenic Signals in Higher Dimensional Spaces

  • Yan YangEmail author
  • Tao Qian
  • Frank Sommen


In the Clifford algebra setting of a Euclidean space on the boundary of a domain it is natural to define a monogenic (analytic) signal to be the boundary value of a monogenic (analytic) function inside the domain. The question is how to define a canonical phase and, correspondingly, a phase derivative. In this paper we give an answer to these questions in the unit ball and in the upper-half space. Among the possible candidates of phases and phase derivatives we decided that the right ones are those that give rise to, as in the one dimensional signal case, the equal relations between the mean of the Fourier frequency and the mean of the phase derivative, and the positivity of the phase derivative of the shifted Cauchy kernel.


Monogenic signals Frequency Poisson kernel Möbius transforms 

Mathematics Subject Classification (2000)

46F15 30G35 


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© Springer Basel AG 2011

Authors and Affiliations

  1. 1.School of Mathematics and Computational ScienceSun Yat-Sen UniversityGuangzhouChina
  2. 2.Department of Mathematics, Faculty of Science and TechnologyUniversity of MacauMacaoChina
  3. 3.Faculty of EngineeringGhent UniversityGhentBelgium

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