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Year 2022, Volume: 4 Issue: 2, 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, Volume: 4 Issue: 2, 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 Volume: 4 Issue: 2

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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