On eigenfunctions of $p$-adic and discrete Fourier transforms

In the paper eigenfunctions and eigenvalues of a $p$-adic Fourier Transform are considered. The completeness and orthonormality of some system of these vectors is proved. These results may be interpreted for the Discrete Fourier Transform (DFT), the fact that allow to determine DFT eigenvectors for some dimensions and to offer a new algorithm as an alternative to the so-called “rapid algorithm”.