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On the calculation of the finite Hankel transform eigenfunctions

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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.

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References

  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)

    MathSciNet  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  3. Aime, C.: Radon approach to shaped and apodized apertures for imaging exoplanets. Astron. Astrophys. 434, 785–794 (2005)

    Article  Google Scholar 

  4. Aime, C.: Apodized apertures for solar coronagraphy. Astron. Astrophys. 467, 317–325 (2007)

    Article  Google Scholar 

  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. Amodio, P., Settanni, G.: A matrix method for the solution of Sturm-Liouville problems. J. Numer. Anal. Ind. Appl. Math. 6, 1–13 (2011)

    MathSciNet  Google Scholar 

  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. Amodio, P., Sgura, I.: High-order finite difference schemes for the solution of second-order BVPs. J. Comput. Appl. Math. 176, 59–76 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    MATH  Google Scholar 

  10. Beskrovny, V.N., Kolobov, M.I.: Quantum-statistical analysis of superresolution for optical systems with circular symmetry. Phys. Rev. A 78, 043824 (2008)

    Article  Google Scholar 

  11. Boivin, R.F.: Eigenfunctions of the Fourier transformation over a circle, part 1: Approximation of Sturmian eigenvalues. DRDC Ottawa Technical Memorandum 342 (2008)

  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. Boyer, G.R.: Pupil filters for moderate superresolution. Appl. Opt. 15, 3089–3093 (1976)

    Article  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  15. Brugnano, L., Trigiante, D.: Solving Differential Problems by Multistep Initial and Boundary Value Methods. Gordon and Breach, Amsterdam (1998)

    Google Scholar 

  16. Coddington, E.A., Levinson, N.: Theory of Differential Equations. McGraw-Hill, New York (1955)

    MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  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. Karoui, A.: Unidimensional and bidimensional prolate spheroidal wave functions and applications. J. Franklin Inst. 348, 1668–1694 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  22. Klug, A., Crowther, R.A.: Three-dimensional image reconstruction from the viewpoint of information theory. Nature 238, 435–440 (1972)

    Article  Google Scholar 

  23. Komarov, I.V., Ponomarev, L.I., Slavyanov, S.Y.: Spheroidal and Coulomb Spheroidal Functions. Nauka, Moscow (1976) (in Russian)

    Google Scholar 

  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. 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. Latham, W.P., Tilton, M.L.: Calculation of prolate functions for optical analysis. Appl. Opt. 26, 2653–2658 (1987)

    Article  Google Scholar 

  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)

    MATH  Google Scholar 

  28. Louis, A.K.: Nonuniqueness in inverse Radon problems: the frequency distribution of the ghosts. Math. Z. 185, 429–440 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  29. Rhodes, D.: On the aperture and pattern space factors for rectangular and circular apertures. IEEE Trans. Antennas Propag. 19, 763–770 (1971)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  32. Slepian, D., Pollak, H.O.: Prolate spheroidal wave functions, Fourier analysis and uncertainty I. Bell Syst. Tech. J. 40, 43–64 (1961)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  Google Scholar 

  34. Weinstein, L.A.: Open Resonators and Open Waveguides. Golem, Boulder (1969)

    Google Scholar 

  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)

    Article  Google Scholar 

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Acknowledgements

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

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Authors and Affiliations

  1. Dipartimento di Matematica, Università di Bari, Via E. Orabona 4, 70125, Bari, Italy

    P. Amodio & G. Settanni

  2. Institute for Computational Mathematics, TU Braunschweig, Pockelsstraße 14, 38106, Brunswick, Germany

    T. Levitina

  3. Institute for Analysis and Scientific Computing, Vienna University of Technology, Wiedner Hauptstrasse 8-10, 1040, Vienna, Austria

    E. B. Weinmüller

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  1. P. Amodio
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  2. T. Levitina
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  3. G. Settanni
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  4. E. B. Weinmüller
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Correspondence to T. Levitina.

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Amodio, P., Levitina, T., Settanni, G. et al. On the calculation of the finite Hankel transform eigenfunctions. J. Appl. Math. Comput. 43, 151–173 (2013). https://doi.org/10.1007/s12190-013-0657-1

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  • DOI: https://doi.org/10.1007/s12190-013-0657-1

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