  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.
