Open AccessMulti-variable quaternionic spectral analysis
Author(s) -
Ilwoo Cho,
Palle E. T. Jørgensen
Publication year - 2021
Publication title -
rocznik akademii górniczo-hutniczej im. stanisława staszica. opuscula mathematica/opuscula mathematica
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.481
H-Index - 16
eISSN - 2300-6919
pISSN - 1232-9274
DOI - 10.7494/opmath.2021.41.3.335
Subject(s) - quaternion , mathematics , ring (chemistry) , combinatorics , functional analysis , variable (mathematics) , spectral analysis , spectral representation , pure mathematics , representation (politics) , functional equation , mathematical analysis , geometry , differential equation , physics , spectroscopy , quantum mechanics , biochemistry , chemistry , organic chemistry , gene , politics , political science , law
In this paper, we consider finite dimensional vector spaces \(\mathbb{H}^n\) over the ring \(\mathbb{H}\) of all quaternions. In particular, we are interested in certain functions acting on \(\mathbb{H}^n\), and corresponding functional equations. Our main results show that (i) all quaternions of \(\mathbb{H}\) are classified by the spectra of their realizations under representation, (ii) all vectors of \(\mathbb{H}^n\) are classified by a canonical extended setting of (i), and (iii) the usual spectral analysis on the matricial ring \(M_n(\mathbb{C})\) of all \((n \times n)\)-matrices over the complex numbers \(\mathbb{C}\) has close connections with certain "non-linear" functional equations on \(\mathbb{H}^n\) up to the classification of (ii).