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

• Published in: 4Open 2019, Vol.2, p.2
• Main Authors: Atakishiyeva, Mesuma K., Atakishiyev, Natig M., Loreto-Hernández, Juan
• Format: Article
• Language: English
• Subjects: Algebra; Algebraic properties; Difference equations; Discrete Fourier transform (DFT); Eigenvalues; Fourier transforms
• ISSN: 2557-0250; 2557-0250
• DOI: 10.1051/fopen/2018010

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 = 2 L .