Advertisement

Discrete Fourier Transform and Theta Function Identities

  • R. A. Malekar
Chapter

Abstract

The classical identities of the Jacobi theta functions are obtained from the multiplicities of the eigenvalues ik and the corresponding eigenvectors of the DFT Φ(n) expressed in terms of the theta functions. An extended version of the classical Watson addition formula and Riemann’s identity on theta functions is derived. Watson addition formula and Riemann’s identity are obtained as a particular case. An extensions of some classical identities corresponding to the theta functions θa,b(x, τ) with a,b\(\frac {1}{3}\mathbb {Z}\) are also derived.

References

  1. 1.
    L. Auslander, R. Tolimieri, Is computing with the finite Fourier transform pure or applied mathematics? Bull. Am. Math. Soc. I, 847–97 (1979)MathSciNetCrossRefGoogle Scholar
  2. 2.
    S. Cooper, P.C. Toh, Determinant identities for theta functions. J. Math. Anal. Appl. 347, 1–7 (2008)MathSciNetCrossRefGoogle Scholar
  3. 3.
    B.W. Dickinson, K. Steiglitz, Eigenvectors and functions of the discrete Fourier transform. Trans. Acoust. Speech Signal Process. 30, 25–31 (1982)MathSciNetCrossRefGoogle Scholar
  4. 4.
    D. Galetti, M.A. Marchiolli, Discrete coherent states and probability distributions in finite dimensional spaces. Ann. Phys. 249, 454–480 (1996)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Z.-G. Liu, Residue theorem and theta function identities. Ramanujan J. 5, 129–151, 389–406 (2001)MathSciNetGoogle Scholar
  6. 6.
    Z.-G. Liu, An addition formula for the Jacobian theta function and its applications. Adv. Math. 212, 389–406 (2007)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Z.-G. Liu, Some inverse relations and theta function identities. Int. J. Number Theory 8(8), 1977–2002 (2012)MathSciNetCrossRefGoogle Scholar
  8. 8.
    R.A. Malekar, H. Bhate, Discrete Fourier transform and Jacobi theta functions identities. J. Math Phys. 51, 023511 (2010)MathSciNetCrossRefGoogle Scholar
  9. 9.
    R.A. Malekar, H. Bhate, Discrete Fourier transform and Riemann’s identities for theta functions. Appl. Math. Lett. 25, 1415–1419 (2012)MathSciNetCrossRefGoogle Scholar
  10. 10.
    V.B. Matveev, Intertwining relations between Fourier transform and discrete Fourier transform, the related functional identities and beyond. Inverse Prob. 17, 633–657 (2001)MathSciNetCrossRefGoogle Scholar
  11. 11.
    J.H. McClennan, T.W. Parks, Eigenvalue and eigenvector decomposition of the discrete Fourier transform. Trans. Acoust. Speech Signal Process. 30, 66–74 (1982)Google Scholar
  12. 12.
    H. Mckean, V. Moll, Elliptic Curves (Cambridge University Press, Cambridge, 1999)zbMATHGoogle Scholar
  13. 13.
    M.L. Mehta, Eigen values and eigenvectors of finite Fourier transform. J Math. Phys. 28(4), 781 (1987)Google Scholar
  14. 14.
    D. Mumford, Tata Lectures on Theta I (Birkhauser, Basel, 1983)CrossRefGoogle Scholar
  15. 15.
    M. Ruzzi, Jacobi θ functions and discrete Fourier transform. J Math. Phys. 47, 063507 (2006)MathSciNetCrossRefGoogle Scholar
  16. 16.
    P.W. Shore, Polynomial time algorithm for prime factorisation and discrete algorithms on quantum computers. SIAM J. Comput. 26, 1484–509 (1997)MathSciNetCrossRefGoogle Scholar
  17. 17.
    H.M. Srivastava, M.P. Choudhary, S. Chaudhary, Some theta-function identities related to Jacobi’s triple-product identity. Eur. J. Pure Appl. Math. 11(1), 1–9 (2018)MathSciNetCrossRefGoogle Scholar
  18. 18.
    R. Tolimieri, The construction of orthonormal bases diagonalising the DFT. Adv. Appl. Math. 5, 56–86 (1984)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Z.X. Wang, D.R. Guo, Special Functions (World Scientific, Singapore, 1989)CrossRefGoogle Scholar
  20. 20.
    E.T. Whittaker, G.N. Watson, A Course of Modern Analysis, 4th edn. (Cambridge University Press, Cambridge, 1927)zbMATHGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • R. A. Malekar
    • 1
  1. 1.Department of MathematicsNational Defence AcademyKhadakwasla, PuneIndia

Personalised recommendations