Open AccessReal Harmonic Analysis on the Special Orthogonal Group
Author(s) -
Taeyoung Lee
Publication year - 2022
Publication title -
international journal of analysis and applications
Language(s) - English
Resource type - Journals
ISSN - 2291-8639
DOI - 10.28924/2291-8639-20-2022-21
Subject(s) - harmonic analysis , group (periodic table) , harmonics , orthogonal functions , mathematics , orthogonal group , noncommutative geometry , unitary state , spherical harmonics , algebra over a field , unitary group , software , set (abstract data type) , harmonic , pure mathematics , computer science , mathematical analysis , programming language , law , chemistry , physics , organic chemistry , quantum mechanics , voltage , political science
This paper presents theoretical analysis and software implementation for real harmonics analysis on the special orthogonal group. Noncommutative harmonic analysis for complex-valued functions on the special orthogonal group has been studied extensively. However, it is customary to treat real harmonic analysis as a special case of complex harmonic analysis, and there have been limited results developed specifically for real-valued functions. Here, we develop a set of explicit formulas for real-valued irreducible unitary representations on the special orthogonal group, and provide several operational properties, such as derivatives, sampling, and Clebsch-Gordon coefficients. Furthermore, we implement both of complex and real harmonics analysis on the special orthogonal group into an open source software package that utilizes parallel processing through the OpenMP library. The efficacy of the presented results are illustrated by benchmark studies and an application to spherical shape matching.