Volume 2, 2019
Difference & Differential Equations and Applications
|Number of page(s)||11|
|Section||Mathematics - Applied Mathematics|
|Published online||04 March 2019|
More on algebraic properties of the discrete Fourier transform raising and lowering operators★
Centro de Investigación en Ciencias, Universidad Autónoma del Estado de Morelos,
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: firstname.lastname@example.org
Accepted: 27 December 2018
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.
Key words: Algebraic properties / Discrete Fourier transform (DFT) / Difference equations
© M.K. Atakishiyeva et al., Published by EDP Sciences 2019
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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