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Year 2022, , 55 - 65, 30.10.2022
https://doi.org/10.47087/mjm.1173367

Abstract

References

  • L.D.Landau, E.M.Lifshitz, Quantum Mechanics (Non-relativistic Theory), Pergamon Press, Oxford, 1991.
  • J.H.McClellan and T.W.Parks, Eigenvalue and eigenvector decomposition of the discrete Fourier transform, IEEE Trans. Audio Electroac., vol. AU-20, 66--74, 1972.
  • L.Auslander and R.Tolimieri, Is computing with the finite Fourier transform pure or applied mathematics? Bull. Amer. Math. Soc., vol. 1, 847--897, 1979.
  • B.W.Dickinson and K.Steiglitz, Eigenvectors and functions of the discrete Fourier transform, IEEE Trans. Acoust. Speech, vol. 30, 25--31, 1982.
  • M.L.Mehta, Eigenvalues and eigenvectors of the finite Fourier transform, J. Math. Phys., vol. 28, 781--785, 1987.
  • V.B.Matveev, Intertwining relations between the Fourier transform and discrete Fourier transform, the related functional identities and beyond, Inverse Prob., vol. 17, 633--657, 2001.
  • N.M.Atakishiyev, On q-extensions of Mehta's eigenvectors of the finite Fourier transform, Int. J. Mod. Phys. A, vol. 21, 4993--5006, 2006.
  • R.A.Horn, C.R.Johnson, Matrix analysis, Cambridge University Press, Cambridge, 2009.
  • M.K.Atakishiyeva and N.M.Atakishiyev, On the raising and lowering difference operators for eigenvectors of the finite Fourier transform, J. Phys: Conf. Ser., vol. 597, 012012, 2015.
  • M.K.Atakishiyeva and N.M.Atakishiyev, On algebraic properties of the discrete raising and lowering operators, associated with the N-dimensional discrete Fourier transform, Adv. Dyn. Syst. Appl., vol. 11, 81--92, 2016.
  • M.K.Atakishiyeva, N.M.Atakishiyev and J.Loreto-Hernández, More on algebraic properties of the discrete Fourier transform raising and lowering operators, 4 Open, vol. 2, 1--11, 2019.
  • M.K.Atakishiyeva, N.M.Atakishiyev and A.Zhedanov, An algebraic interpretation of the intertwining operators associated with the discrete Fourier transform, J. Math. Phys., vol. 62, 101704, 2021.
  • A.S.Zhedanov, "Hidden symmetry" of Askey-Wilson polynomials, Theoretical and Mathematical Physics, vol. 89, 1146--1157, 1991.
  • P.Terwilliger, The Universal Askey-Wilson Algebra, SIGMA, vol. 7, 069, 2011.
  • T.H.Koornwinder, The relationship between Zhedanov's algebra AW(3) and the double affine Hecke algebra in the rank one case, SIGMA, vol. 3, 063, 2007, 15 pp.
  • T.H.Koornwinder, Zhedanov's algebra AW(3) and the double affine Hecke algebra in the rank one case.II. The spherical subalgebra, SIGMA, vol. 4, 052, 2008, 17 pp.
  • P. Baseilhac, S. Tsujimoto, L. Vinet, and A. Zhedanov, The Heun-Askey-Wilson Algebra and the Heun Operator of Askey-Wilson Type, Annales Henri Poincaré, vol. 20, 3091--3112, 2019.
  • M.K.Atakishiyeva, N.M.Atakishiyev and J.Méndez Franco, On a discrete number operator associated with the 5D discrete Fourier transform, Springer Proceedings in Mathematics & Statistics, vol. 164, 273--292, 2016.
  • M.C.Pereyra and L.A.Ward, Harmonic analysis: from Fourier to wavelets, AMS, Providence, Rhode Island, 2012.
  • K.R.Rao, D.N.Kim, J.J.Hwang, Fast Fourier Transform: Algorithms and Applications, Springer, Dordrecht, 2010.

The eigenvalues and eigenvectors of the 5D discrete Fourier transform number operator revisited

Year 2022, , 55 - 65, 30.10.2022
https://doi.org/10.47087/mjm.1173367

Abstract

A systematic analytic approach to the evaluation of the eigenvalues and eigenvectors of the 5D discrete number operator N_5 is formulated. This approach is essentially based on the use of the symmetricity of 5D discrete Fourier transform operator fi_5 with respect to the discrete reflection operator P_d.

References

  • L.D.Landau, E.M.Lifshitz, Quantum Mechanics (Non-relativistic Theory), Pergamon Press, Oxford, 1991.
  • J.H.McClellan and T.W.Parks, Eigenvalue and eigenvector decomposition of the discrete Fourier transform, IEEE Trans. Audio Electroac., vol. AU-20, 66--74, 1972.
  • L.Auslander and R.Tolimieri, Is computing with the finite Fourier transform pure or applied mathematics? Bull. Amer. Math. Soc., vol. 1, 847--897, 1979.
  • B.W.Dickinson and K.Steiglitz, Eigenvectors and functions of the discrete Fourier transform, IEEE Trans. Acoust. Speech, vol. 30, 25--31, 1982.
  • M.L.Mehta, Eigenvalues and eigenvectors of the finite Fourier transform, J. Math. Phys., vol. 28, 781--785, 1987.
  • V.B.Matveev, Intertwining relations between the Fourier transform and discrete Fourier transform, the related functional identities and beyond, Inverse Prob., vol. 17, 633--657, 2001.
  • N.M.Atakishiyev, On q-extensions of Mehta's eigenvectors of the finite Fourier transform, Int. J. Mod. Phys. A, vol. 21, 4993--5006, 2006.
  • R.A.Horn, C.R.Johnson, Matrix analysis, Cambridge University Press, Cambridge, 2009.
  • M.K.Atakishiyeva and N.M.Atakishiyev, On the raising and lowering difference operators for eigenvectors of the finite Fourier transform, J. Phys: Conf. Ser., vol. 597, 012012, 2015.
  • M.K.Atakishiyeva and N.M.Atakishiyev, On algebraic properties of the discrete raising and lowering operators, associated with the N-dimensional discrete Fourier transform, Adv. Dyn. Syst. Appl., vol. 11, 81--92, 2016.
  • M.K.Atakishiyeva, N.M.Atakishiyev and J.Loreto-Hernández, More on algebraic properties of the discrete Fourier transform raising and lowering operators, 4 Open, vol. 2, 1--11, 2019.
  • M.K.Atakishiyeva, N.M.Atakishiyev and A.Zhedanov, An algebraic interpretation of the intertwining operators associated with the discrete Fourier transform, J. Math. Phys., vol. 62, 101704, 2021.
  • A.S.Zhedanov, "Hidden symmetry" of Askey-Wilson polynomials, Theoretical and Mathematical Physics, vol. 89, 1146--1157, 1991.
  • P.Terwilliger, The Universal Askey-Wilson Algebra, SIGMA, vol. 7, 069, 2011.
  • T.H.Koornwinder, The relationship between Zhedanov's algebra AW(3) and the double affine Hecke algebra in the rank one case, SIGMA, vol. 3, 063, 2007, 15 pp.
  • T.H.Koornwinder, Zhedanov's algebra AW(3) and the double affine Hecke algebra in the rank one case.II. The spherical subalgebra, SIGMA, vol. 4, 052, 2008, 17 pp.
  • P. Baseilhac, S. Tsujimoto, L. Vinet, and A. Zhedanov, The Heun-Askey-Wilson Algebra and the Heun Operator of Askey-Wilson Type, Annales Henri Poincaré, vol. 20, 3091--3112, 2019.
  • M.K.Atakishiyeva, N.M.Atakishiyev and J.Méndez Franco, On a discrete number operator associated with the 5D discrete Fourier transform, Springer Proceedings in Mathematics & Statistics, vol. 164, 273--292, 2016.
  • M.C.Pereyra and L.A.Ward, Harmonic analysis: from Fourier to wavelets, AMS, Providence, Rhode Island, 2012.
  • K.R.Rao, D.N.Kim, J.J.Hwang, Fast Fourier Transform: Algorithms and Applications, Springer, Dordrecht, 2010.
There are 20 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Articles
Authors

Natig Atakishiyev 0000-0002-8115-0574

Publication Date October 30, 2022
Acceptance Date October 28, 2022
Published in Issue Year 2022

Cite

APA Atakishiyev, N. (2022). The eigenvalues and eigenvectors of the 5D discrete Fourier transform number operator revisited. Maltepe Journal of Mathematics, 4(2), 55-65. https://doi.org/10.47087/mjm.1173367
AMA Atakishiyev N. The eigenvalues and eigenvectors of the 5D discrete Fourier transform number operator revisited. Maltepe Journal of Mathematics. October 2022;4(2):55-65. doi:10.47087/mjm.1173367
Chicago Atakishiyev, Natig. “The Eigenvalues and Eigenvectors of the 5D Discrete Fourier Transform Number Operator Revisited”. Maltepe Journal of Mathematics 4, no. 2 (October 2022): 55-65. https://doi.org/10.47087/mjm.1173367.
EndNote Atakishiyev N (October 1, 2022) The eigenvalues and eigenvectors of the 5D discrete Fourier transform number operator revisited. Maltepe Journal of Mathematics 4 2 55–65.
IEEE N. Atakishiyev, “The eigenvalues and eigenvectors of the 5D discrete Fourier transform number operator revisited”, Maltepe Journal of Mathematics, vol. 4, no. 2, pp. 55–65, 2022, doi: 10.47087/mjm.1173367.
ISNAD Atakishiyev, Natig. “The Eigenvalues and Eigenvectors of the 5D Discrete Fourier Transform Number Operator Revisited”. Maltepe Journal of Mathematics 4/2 (October 2022), 55-65. https://doi.org/10.47087/mjm.1173367.
JAMA Atakishiyev N. The eigenvalues and eigenvectors of the 5D discrete Fourier transform number operator revisited. Maltepe Journal of Mathematics. 2022;4:55–65.
MLA Atakishiyev, Natig. “The Eigenvalues and Eigenvectors of the 5D Discrete Fourier Transform Number Operator Revisited”. Maltepe Journal of Mathematics, vol. 4, no. 2, 2022, pp. 55-65, doi:10.47087/mjm.1173367.
Vancouver Atakishiyev N. The eigenvalues and eigenvectors of the 5D discrete Fourier transform number operator revisited. Maltepe Journal of Mathematics. 2022;4(2):55-6.

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