Research Article

More on algebraic properties of the discrete Fourier transform raising and lowering operators^★

¹ Centro de Investigación en Ciencias, Universidad Autónoma del Estado de Morelos, Cuernavaca, 62250 Morelos, México
² Instituto de Matemáticas, Unidad Cuernavaca, Universidad Nacional Autónoma de México, Cuernavaca, 62210 Morelos, México
³ Facultad de Ciencias, Universidad Autónoma de San Luis Potosí, San Luis Potosí, 78290 SLP, México

^* Corresponding author: natig_atakishiyev@hotmail.com

Received: 20 December 2018
Accepted: 27 December 2018

Abstract

In the present work, we discuss some additional findings concerning algebraic properties of the N-dimensional discrete Fourier transform (DFT) raising and lowering difference operators, recently introduced in [Atakishiyeva MK, Atakishiyev NM (2015), J Phys: Conf Ser 597, 012012; Atakishiyeva MK, Atakishiyev NM (2016), Adv Dyn Syst Appl 11, 81–92]. In particular, we argue that the most authentic symmetrical form of discretization of the integral Fourier transform may be constructed as the discrete Fourier transforms based on the odd points N only, while in the discrete Fourier transforms on the even points N this symmetry is spontaneously broken. This heretofore undetected distinction between odd and even dimensions is shown to be intimately related with the newly revealed algebraic properties of the above-mentioned DFT raising and lowering difference operators and, of course, is very consistent with the well-known formula for the multiplicities of the eigenvalues, associated with the N-dimensional DFT. In addition, we propose a general approach to deriving the eigenvectors of the discrete number operators N^(N), that avoids the above-mentioned pitfalls in the structure of each even-dimensional case N = 2L.