Alternating group and multivariate exponential functions

TL;DR

This paper introduces multivariate exponential functions symmetric under the alternating group, explores their properties as Laplace eigenfunctions, and develops Fourier transforms using these functions as kernels.

Contribution

It defines and analyzes a new class of symmetric multivariate exponential functions related to the alternating group and constructs associated Fourier transforms.

Findings

Functions are eigenfunctions of the Laplace operator.

Fourier transforms are constructed using these functions as kernels.

Eigenfunctions of the integral Fourier transforms are identified.

Abstract

We define and study multivariate exponential functions, symmetric with respect to the alternating group A_n, which is a subgroup of the permutation (symmetric) group S_n. These functions are connected with multivariate exponential functions, determined as the determinants of matrices whose entries are exponential functions of one variable. Our functions are eigenfunctions of the Laplace operator. By means of alternating multivariate exponential functions three types of Fourier transforms are constructed: expansions into corresponding Fourier series, integral Fourier transforms, and multivariate finite Fourier transforms. Alternating multivariate exponential functions are used as a kernel in all these Fourier transforms. Eigenfunctions of the integral Fourier transforms are obtained.