(Anti)symmetric multivariate exponential functions and corresponding Fourier transforms

TL;DR

This paper introduces symmetrized and antisymmetrized multivariate exponential functions, explores their properties as Laplace eigenfunctions, and develops corresponding Fourier transforms, including series, integral, and finite types, with identified eigenfunctions.

Contribution

It defines new symmetric and antisymmetric multivariate exponential functions and establishes their roles as eigenfunctions of associated Fourier transforms, expanding the mathematical framework for multivariate analysis.

Findings

Eigenfunctions of the integral Fourier transforms are identified.

Three types of Fourier transforms are constructed: series, integral, and finite.

Symmetrized and antisymmetrized functions are eigenfunctions of the Laplace operator.

Abstract

We define and study symmetrized and antisymmetrized multivariate exponential functions. They are defined as determinants and antideterminants of matrices whose entries are exponential functions of one variable. These functions are eigenfunctions of the Laplace operator on corresponding fundamental domains satisfying certain boundary conditions. To symmetric and antisymmetric multivariate exponential functions there correspond Fourier transforms. There are three types of such Fourier transforms: expansions into corresponding Fourier series, integral Fourier transforms, and multivariate finite Fourier transforms. Eigenfunctions of the integral Fourier transforms are found.