Three variable exponential functions of the alternating group

TL;DR

This paper introduces a new class of three-variable special functions based on the alternating subgroup of the permutation group, enabling Fourier-like expansion and interpolation of digital data on arbitrary lattices.

Contribution

It presents novel three-variable functions linked to the alternating group, extending Fourier analysis techniques to irregular lattice data.

Findings

Functions enable Fourier-like expansion of lattice data

Connection established with $E$-functions of $C_3$

Continuous interpolation of 3D data demonstrated

Abstract

New class of special functions of three real variables, based on the alternating subgroup of the permutation group $S_3$, is studied. These functions are used for Fourier-like expansion of digital data given on lattice of any density and general position. Such functions have only trivial analogs in one and two variables; a connection to the $E-$functions of $C_3$ is shown. Continuous interpolation of the three dimensional data is studied and exemplified.