Journal of Fourier Analysis and Applications

, Volume 17, Issue 2, pp 240–264 | Cite as

Sampling Theorem and Discrete Fourier Transform on the Hyperboloid

Article

Abstract

Using Coherent-State (CS) techniques, we prove a sampling theorem for holomorphic functions on the hyperboloid (or its stereographic projection onto the open unit disk \(\mathbb{D}_{1}\)), seen as a homogeneous space of the pseudo-unitary group SU(1,1). We provide a reconstruction formula for bandlimited functions, through a sinc-type kernel, and a discrete Fourier transform from N samples properly chosen. We also study the case of undersampling of band-unlimited functions and the conditions under which a partial reconstruction from N samples is still possible and the accuracy of the approximation, which tends to be exact in the limit N→∞.

Keywords

Holomorphic functions Coherent states Discrete Fourier transform Sampling Discrete frames 

Mathematics Subject Classification (2000)

32A10 42B05 94A12 94A20 81R30 

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References

  1. 1.
    Ali, S.T., Antoine, J.-P., Gazeau, J.P.: Coherent States, Wavelets and Their Generalizations. Springer, Berlin (2000) MATHGoogle Scholar
  2. 2.
    Antoine, J.-P., Hohouto, A.L.: Discrete frames of Poincaré coherent states in 1+3 dimensions. J. Fourier Anal. Appl. 9, 141–173 (2003) MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Ben-Israel, A., Greville, T.N.E.: Generalized Inverses. Springer, Berlin (2003) MATHGoogle Scholar
  4. 4.
    Calixto, M., Guerrero, J.: Wavelet transform on the circle and the real line: a unified group-theoretical treatment. Appl. Comput. Harmon. Anal. 21, 204–229 (2006) MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Calixto, M., Guerrero, J., Sánchez-Monreal, J.C.: Sampling theorem and discrete Fourier transform on the Riemann sphere. J. Fourier Anal. Appl. 14, 538–567 (2008) MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Calixto, M., Guerrero, J., Sánchez-Monreal, J.C.: Sampling theorems and discrete Fourier transforms on curved phase spaces of constant curvature: a unified treatment. In progress Google Scholar
  7. 7.
    Chirikjian, G.S., Kyatkin, A.: Engineering Applications of Noncommutative Harmonic Analysis. CRC Press, Boca Raton (2001) MATHGoogle Scholar
  8. 8.
    Christensen, O.: An Introduction to Frames and Riesz Bases. Birkhäuser, Boston (2003) MATHGoogle Scholar
  9. 9.
    Daoud, M., Jellal, A.: Quantum Hall droplets on disk and effective Weiss-Zumino-Witten action for edge states. Int. J. Geom. Methods Mod. Phys. 4, 1187–1204 (2007) MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Davis, P.J.: Circulant Matrices. Chelsea, New York (1994) MATHGoogle Scholar
  11. 11.
    Ebata, M., Eguchi, M., Koizumi, S., Kumahara, K.: Analogues of sampling theorems for some homogeneous spaces. Hiroshima Math. J. 36, 125–140 (2006) MathSciNetMATHGoogle Scholar
  12. 12.
    Ebata, M., Eguchi, M., Koizumi, S., Kumahara, K.: On sampling formulas on symmetric spaces. J. Fourier Anal. Appl. 12, 1–15 (2006) MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Feichtinger, H., Pesenson, I.: A reconstruction method for band-limited signals on the hyperbolic plane. Sampl. Theory Signal Image Process. 4, 107–119 (2005) MathSciNetMATHGoogle Scholar
  14. 14.
    Führ, H.: Abstract Harmonic Analysis of Continuous Wavelet Transforms. Springer, Berlin (2005) MATHGoogle Scholar
  15. 15.
    Grossmann, A., Morlet, J., Paul, T.: Transforms associated to square integrable group representations I. General results. J. Math. Phys. 26, 2473–2479 (1985) MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    Guerrero, J., Aldaya, V.: Invariant measures on polarized submanifolds in group quantization. J. Math. Phys. 41, 6747–6765 (2000) MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    Holschneider, M.: Wavelets: an Analysis Tool. Oxford University Press, London (1998) MATHGoogle Scholar
  18. 18.
    Klauder, J.R., Skagerstam, Bo-Sture: Coherent States: Applications in Physics and Mathematical Physics. Singapore, World Scientific (1985) MATHGoogle Scholar
  19. 19.
    Kyatkin, A., Chirikjian, G.S.: Computation of robot configuration and workspaces via the Fourier transform on the discrete motion-group. Int. J. Robot. Res. 18 601–615 (1999) CrossRefGoogle Scholar
  20. 20.
    Kyatkin, A., Chirikjian, G.S.: Algorithms for fast convolutions on motion groups. Appl. Comput. Harmon. Anal. 9, 220–241 (2000) MathSciNetMATHCrossRefGoogle Scholar
  21. 21.
    Maslen, D.: Efficient computation of Fourier transforms on compact groups. J. Fourier Anal. Appl. 4, 19–52 (1998) MathSciNetMATHCrossRefGoogle Scholar
  22. 22.
    Maslen, D.: Sampling of functions and sections for compact Groups. Mod. Signal Process. 46, 247–280 (2003) MathSciNetGoogle Scholar
  23. 23.
    Perelomov, A.: Generalized Coherent States and Their Applications. Springer, Berlin (1986) MATHGoogle Scholar
  24. 24.
    Pesenson, I.: A sampling theorem on homogeneous manifolds. Trans. Am. Math. Soc. 352, 4257–4269 (2000) MathSciNetMATHCrossRefGoogle Scholar
  25. 25.
    Pesenson, I.: Poincaré-type inequalities and reconstruction of Paley-Wiener functions on manifolds. J. Geom. Anal. 14, 101–121 (2004) MathSciNetMATHGoogle Scholar
  26. 26.
    Pesenson, I.: Deconvolution of band limited functions on non-compact symmetric spaces. Houst. J. Math. 32, 183–204 (2006) MathSciNetMATHGoogle Scholar
  27. 27.
    Pesenson, I.: Paley-Wiener approximations and multiscale approximations in Sobolev and Besov spaces on manifolds. J. Geom. Anal. 19, 390–419 (2009) MathSciNetMATHCrossRefGoogle Scholar
  28. 28.
    Stenzel, M.B.: A reconstruction theorem for Riemannian symmetric spaces of noncompact type. J. Fourier Anal. Appl. 15, 839–856 (2009) MathSciNetMATHCrossRefGoogle Scholar
  29. 29.
    Wigner, E.P., Group Theory and its Applications to the Quantum Mechanics of Atomic Spectra. Academic Press, New York (1959) Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  • M. Calixto
    • 1
  • J. Guerrero
    • 2
  • J. C. Sánchez-Monreal
    • 1
  1. 1.Departamento de Matemática Aplicada y EstadísticaUniversidad Politécnica de CartagenaCartagenaSpain
  2. 2.Departamento de Matemática Aplicada, Facultad de InformáticaUniversidad de MurciaMurciaSpain

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