The main objective of this article is to give a survey on elementary functions in the context of quaternionic analysis. We define some of their more common properties, which as in the real and complex cases, will be familiar to the reader. This leads to the consideration of quaternion-valued functions depending on a quaternion variable, that is, functions whose input and output are quaternions....

The recently introduced generalized Hamiltonian–Real (GHR) calculus comprises, for the first time, the product and chain rules that makes it a powerful tool for quaternion-based optimization and adaptive signal processing. In this paper, we introduce novel dual relationships between the GHR calculus and multivariate real calculus, in order to provide a new, simpler proof of the GHR derivative r...

In this talk we introduce and study quaternion-valued positive definite functions on locally compact Abelian groups, real countably Hilbertian nuclear spaces and on the space of countably infinite tuples of real numbers endowed with the Tychonoff topology. In particular, we prove a quaternionic version of the Bochner-Minlos theorem. We will see that in all these various settings the integral re...

This paper introduces some construction methods for amicable orthogonal designs over the real and quaternion domain which have not been explored for code design before. The proposed construction methods generate a large number of amicable orthogonal designs of quaternions. It is also shown that amicable orthogonal designs of quaternions can be used to construct restricted quaternion orthogonal ...

A variety of universal similarity factorization equalities over real Clifford algebras Rp,q are established. On the basis of these equalities, real, complex and quaternion matrix representations of elements in Rp,q can be explicitly determined.