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Journal of Applied Mathematics and Computing

, Volume 43, Issue 1–2, pp 151–173 | Cite as

On the calculation of the finite Hankel transform eigenfunctions

  • P. Amodio
  • T. Levitina
  • G. Settanni
  • E. B. Weinmüller
Original Research

Abstract

In this work, we discuss the numerical computation of the eigenvalues and eigenfunctions of the finite (truncated) Hankel transform, important for numerous applications. Due to the very special behavior of the Hankel transform eigenfunctions, their direct numerical calculation often causes an essential loss of accuracy.

Here, we present several simple, efficient and robust numerical techniques to compute Hankel transform eigenfunctions via the associated singular self-adjoint Sturm-Liouville operator. The properties of the proposed approaches are compared and illustrated by means of numerical experiments.

Keywords

Finite Hankel transform Generalized spheroidal wave functions Singular Sturm-Liouville problem Finite difference schemes Prüfer angle 

Mathematics Subject Classification (2000)

34B16 34B24 34L16 65F15 65L10 65L15 65R10 65R20 

Notes

Acknowledgements

The authors are very grateful for the warm hospitality of Vienna University of Technology while working on this paper.

References

  1. 1.
    Abramov, A.A., Dyshko, A.L., Konyukhova, N.B., Pak, T.V., Pariiskii, B.S.: Evaluation of prolate spheroidal function by solving the corresponding differential equations. USSR Comput. Math. Math. Phys. 24, 1–11 (1984) MathSciNetGoogle Scholar
  2. 2.
    Abramov, A.A., Dyshko, A.L., Konyukhova, N.B., Levitina, T.V.: Computation of radial wave functions for spheroids and triaxial ellipsoids by the modified phase function method. Comput. Math. Math. Phys. 31, 25–42 (1991) MathSciNetzbMATHGoogle Scholar
  3. 3.
    Aime, C.: Radon approach to shaped and apodized apertures for imaging exoplanets. Astron. Astrophys. 434, 785–794 (2005) CrossRefGoogle Scholar
  4. 4.
    Aime, C.: Apodized apertures for solar coronagraphy. Astron. Astrophys. 467, 317–325 (2007) CrossRefGoogle Scholar
  5. 5.
    Amodio, P., Settanni, G.: High order finite difference schemes for the numerical solution of eigenvalue problems for IVPs in ODEs. In: Numerical Analysis and Applied Mathematics. AIP Conf. Proc., vol. 1281, pp. 202–205 (2010) Google Scholar
  6. 6.
    Amodio, P., Settanni, G.: A matrix method for the solution of Sturm-Liouville problems. J. Numer. Anal. Ind. Appl. Math. 6, 1–13 (2011) MathSciNetGoogle Scholar
  7. 7.
    Amodio, P., Settanni, G.: A stepsize variation strategy for the solution of regular Sturm-Liouville problems. In: Numerical Analysis and Applied Mathematics. AIP Conf. Proc., vol. 1389, pp. 1335–1338 (2011) Google Scholar
  8. 8.
    Amodio, P., Sgura, I.: High-order finite difference schemes for the solution of second-order BVPs. J. Comput. Appl. Math. 176, 59–76 (2005) MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Auzinger, W., Karner, E., Koch, O., Weinmüller, E.B.: Collocation methods for the solution of eigenvalue problems for singular ordinary differential equations. Opusc. Math. 26, 229–241 (2006) zbMATHGoogle Scholar
  10. 10.
    Beskrovny, V.N., Kolobov, M.I.: Quantum-statistical analysis of superresolution for optical systems with circular symmetry. Phys. Rev. A 78, 043824 (2008) CrossRefGoogle Scholar
  11. 11.
    Boivin, R.F.: Eigenfunctions of the Fourier transformation over a circle, part 1: Approximation of Sturmian eigenvalues. DRDC Ottawa Technical Memorandum 342 (2008) Google Scholar
  12. 12.
    Borgiotti, G.: Hyperspheroidal functions—high beam efficiency illumination for circular antennas. In: Antennas and Propagation Society International Symposium, pp. 30–39 (1969) Google Scholar
  13. 13.
    Boyer, G.R.: Pupil filters for moderate superresolution. Appl. Opt. 15, 3089–3093 (1976) CrossRefGoogle Scholar
  14. 14.
    Brander, O., De Facio, B.: The role of filters and the singular-value decomposition for the inverse Born approximation. Inverse Probl. 2, 375–393 (1986) CrossRefzbMATHGoogle Scholar
  15. 15.
    Brugnano, L., Trigiante, D.: Solving Differential Problems by Multistep Initial and Boundary Value Methods. Gordon and Breach, Amsterdam (1998) Google Scholar
  16. 16.
    Coddington, E.A., Levinson, N.: Theory of Differential Equations. McGraw-Hill, New York (1955) zbMATHGoogle Scholar
  17. 17.
    de Hoog, F.R., Weiss, R.: Difference methods for boundary value problems with a singularity of the first kind. SIAM J. Numer. Anal. 13, 775–813 (1976) MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Hammerling, R., Koch, O., Simon, C., Weinmüller, E.B.: Numerical solution of singular ODE eigenvalue problems in electronic structure computations. J. Comput. Phys. 181, 1557–1561 (2010) CrossRefzbMATHGoogle Scholar
  19. 19.
    Heurtley, J.C.: Hyperspheroidal functions—optical resonators with circular mirrors. In: Quasi-Optics, Proc. Symposium on Quasi-Optics, New York, June 8–10, 1964. Microwave Res. Inst. Symp. Ser., vol. 14, p. 367. Polytechnic Press, Brooklyn (1964) Google Scholar
  20. 20.
    Karoui, A.: Unidimensional and bidimensional prolate spheroidal wave functions and applications. J. Franklin Inst. 348, 1668–1694 (2011) MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Karoui, A., Moumni, T.: Spectral analysis of the finite Hankel transform and circular prolate spheroidal wave functions. J. Comput. Appl. Math. 233, 315–333 (2009) MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Klug, A., Crowther, R.A.: Three-dimensional image reconstruction from the viewpoint of information theory. Nature 238, 435–440 (1972) CrossRefGoogle Scholar
  23. 23.
    Komarov, I.V., Ponomarev, L.I., Slavyanov, S.Y.: Spheroidal and Coulomb Spheroidal Functions. Nauka, Moscow (1976) (in Russian) Google Scholar
  24. 24.
    Kuznetsov, N.V.: On eigen-functions of an integral equation. In: Mathematical Problems in the Theory of Wave Propagation, Part 3. Zap. Nauchn. Sem. LOMI, pp. 66–150. Nauka, Leningrad (1970) Google Scholar
  25. 25.
    Larsson, B., Levitina, T.V., Brändas, E.J.: On generalized prolate spheroidal functions. In: Proc. CMMSE-2002, Alicante, vol. II, pp. 220–223 (2002) Google Scholar
  26. 26.
    Latham, W.P., Tilton, M.L.: Calculation of prolate functions for optical analysis. Appl. Opt. 26, 2653–2658 (1987) CrossRefGoogle Scholar
  27. 27.
    Levitina, T.V., Brändas, E.J.: Computational techniques for prolate spheroidal wave functions in signal processing. J. Comput. Methods Sci. Eng. 1, 287–313 (2001) zbMATHGoogle Scholar
  28. 28.
    Louis, A.K.: Nonuniqueness in inverse Radon problems: the frequency distribution of the ghosts. Math. Z. 185, 429–440 (1984) MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Rhodes, D.: On the aperture and pattern space factors for rectangular and circular apertures. IEEE Trans. Antennas Propag. 19, 763–770 (1971) CrossRefGoogle Scholar
  30. 30.
    Sherif, S.S., Foreman, M.R., Török, P.: Eigenfunction expansion of the electric fields in the focal region of a high numerical aperture focusing system. Opt. Express 16, 3397–3407 (2008) CrossRefGoogle Scholar
  31. 31.
    Slepian, D.: Prolate spheroidal wave functions, Fourier analysis and uncertainty, IV: Extensions to many dimensions; generalized prolate spheroidal functions. Bell Syst. Tech. J. 43, 3009–3058 (1964) MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Slepian, D., Pollak, H.O.: Prolate spheroidal wave functions, Fourier analysis and uncertainty I. Bell Syst. Tech. J. 40, 43–64 (1961) MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Tam, K.C., Perez-Mendez, V., MacDonald, B.: Limited angle 3-D reconstructions from continuous and pinhole projections. IEEE Trans. Nucl. Sci. 27, 445–458 (1980) CrossRefGoogle Scholar
  34. 34.
    Weinstein, L.A.: Open Resonators and Open Waveguides. Golem, Boulder (1969) Google Scholar
  35. 35.
    Zhang, X.: Wavenumber spectrum of very short wind waves: an application of two-dimensional Slepian windows to spectral estimation. J. Atmos. Ocean. Technol. 11, 489–505 (1994) CrossRefGoogle Scholar

Copyright information

© Korean Society for Computational and Applied Mathematics 2013

Authors and Affiliations

  • P. Amodio
    • 1
  • T. Levitina
    • 2
  • G. Settanni
    • 1
  • E. B. Weinmüller
    • 3
  1. 1.Dipartimento di MatematicaUniversità di BariBariItaly
  2. 2.Institute for Computational MathematicsTU BraunschweigBrunswickGermany
  3. 3.Institute for Analysis and Scientific ComputingVienna University of TechnologyViennaAustria

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