Three dimensional C-, S- and E-transforms

TL;DR

This paper introduces three-dimensional Fourier-like transforms based on Lie groups, detailing their properties, orthogonality, and function expansions on continuous regions and lattice grids.

Contribution

It presents new 3D Fourier-like transforms derived from Lie groups, with detailed analysis of their orthogonality and function expansion properties.

Findings

Orthogonality of functions within each family established

Discrete and continuous orthogonality properties described

Framework for expanding functions on 3D regions and lattices provided

Abstract

Three dimensional continuous and discrete Fourier-like transforms, based on the three simple and four semisimple compact Lie groups of rank 3, are presented. For each simple Lie group, there are three families of special functions ($C$-, $S$-, and $E$-functions) on which the transforms are built. Pertinent properties of the functions are described in detail, such as their orthogonality within each family, when integrated over a finite region $F$ of the 3-dimensional Euclidean space (continuous orthogonality), as well as when summed up over a lattice grid $F_M\subset F$ (discrete orthogonality). The positive integer $M$ sets up the density of the lattice containing $F_M$. The expansion of functions given either on $F$ or on $F_M$ is the paper's main focus.