Kit Ian Kou — Page 4

arXiv2016

Generalized Sampling Expansions Associated with Quaternion Fourier Transform

Quaternion-valued signals along with quaternion Fourier transforms (QFT)provide an effective framework for vector-valued signal and image processing. However, the sampling theory of quaternion valued signals has not been well developed. In this paper, we present the generalized sampling expansions associated with QFT by using the generalized translation and convolution. We show that a σ-bandlimited quaternion valued signal in QFT sense can be reconstructed from the samples of output signals of M linear systems based on QFT. Quaternion linear canonical transform (QLCT) is a generalization of QFT with six parameters. Using the relationship between QFT, we derive the sampling formula for σ-bandlimited quaternion-valued signal in QLCT sense. Examples are given to illustrate our results.

math.FA↗

arXiv2009

Hilbert transforms and the Cauchy integral in euclidean space

We generalize the notion of harmonic conjugate functions and Hilbert transforms to higher dimensional euclidean spaces, in the setting of differential forms and the Hodge-Dirac system. These conjugate functions are in general far from being unique, but under suitable boundary conditions we prove existence and uniqueness of conjugates. The proof also yields invertibility results for a new class of generalized double layer potential operators on Lipschitz surfaces and boundedness of related Hilbert transforms.

math.AP↗