4 open (Jan 2019)

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

  • Atakishiyeva Mesuma K.,
  • Atakishiyev Natig M.,
  • Loreto-Hernández Juan

DOI
https://doi.org/10.1051/fopen/2018010
Journal volume & issue
Vol. 2
p. 2

Abstract

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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) N(N) , that avoids the above-mentioned pitfalls in the structure of each even-dimensional case N = 2L.

Keywords