@Article{Cho2021,
author = "Cho, Ilwoo and Jorgensen, Palle E.T.",
title = "Multi-variable quaternionic spectral analysis",
journal = "Opuscula Math.",
year = "2021",
volume = "41",
number = "3",
pages = "335--379",
doi = "https://doi.org/10.7494/OpMath.2021.41.3.335",
url = "https://doi.org/10.7494/OpMath.2021.41.3.335",
abstract = "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).",
keywords = "the quaternions \(\mathbb{H}\), vector spaces \(\mathbb{H}^n\) over \(\mathbb{H}\), \(q\)-spectral forms, \(q\)-spectral functions",
issn = "1232-9274"
}