Hilbert transforms in clifford analysis

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 and 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 particularly suited for a treatment of multidimensional phenomena since all dimensions are encompassed at once as an intrinsic feature. © 2010 Springer-Verlag London Limited.