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Multi-variable quaternionic spectral analysis
Journal article   Open access   Peer reviewed

Multi-variable quaternionic spectral analysis

Ilwoo Cho and Palle E.T. Jorgensen
Rocznik Akademii Górniczo-Hutniczej im. Stanisława Staszica. Opuscula Mathematica, Vol.41(3), pp.335-379
04/01/2021
DOI: 10.7494/OpMath.2021.41.3.335
url
https://doi.org/10.7494/OpMath.2021.41.3.335View
Published (Version of record) Open Access

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).
 (q\)-spectral forms (q\)-spectral functions the quaternions \(\mathbb{h}\) vector spaces \(\mathbb{h}^n\) over \(\mathbb{h}\)

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