(Anti)symmetric multivariate trigonometric functions and corresponding Fourier transforms

TL;DR

This paper introduces symmetric and antisymmetric multivariate sine and cosine functions as eigenfunctions of the Laplace operator, establishing their orthogonality and defining new Fourier-like transforms in multiple dimensions.

Contribution

It presents novel multivariate special functions based on determinants and antideterminants, and develops their orthogonality properties and Fourier transforms.

Findings

Functions are eigenfunctions of the Laplace operator.

Orthogonality of functions enables Fourier-like transforms.

New multivariate transforms generalize classical Fourier analysis.

Abstract

Four families of special functions, depending on n variables, are studied. We call them symmetric and antisymmetric multivariate sine and cosine functions. They are given as determinants or antideterminants of matrices, whose matrix elements are sine or cosine functions of one variable each. These functions are eigenfunctions of the Laplace operator, satisfying specific conditions at the boundary of a certain domain F of the n-dimensional Euclidean space. Discrete and continuous orthogonality on F of the functions within each family, allows one to introduce symmetrized and antisymmetrized multivariate Fourier-like transforms, involving the symmetric and antisymmetric multivariate sine and cosine functions.