Skip to main content
Log in

Uniform Approximation and Explicit Estimates for the Prolate Spheroidal Wave Functions

  • Published:
Constructive Approximation Aims and scope

Abstract

For fixed c, prolate spheroidal wave functions (PSWFs), denoted by \(\psi _{n, c}\), form an orthogonal basis with remarkable properties for the space of band-limited functions with bandwith c. They have been widely studied and used after the seminal work of D. Slepian and his co-authors. In several applications, uniform estimates of the \(\psi _{n,c}\) in n and c are needed. To progress in this direction, we push forward the uniform approximation error bounds and give an explicit approximation of their values at 1 in terms of the Legendre complete elliptic integral of the first kind. Also, we give an explicit formula for the accurate approximation of the eigenvalues of the Sturm–Liouville operator associated with the PSWFs.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions. Dover Publication, New York (1972)

    MATH  Google Scholar 

  2. Andrews, G.E., Askey, R., Roy, R.: Special Functions. Cambridge University Press, Cambridge (2000)

    MATH  Google Scholar 

  3. Bonami, A., Karoui, A.: Uniform bounds of prolate spheroidal wave functions and eigenvalues decay. C. R. Math. Acad. Sci. Paris 352, 229–234 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  4. Bonami, A., Karoui, A.: Spectral decay of time and frequency limiting operator. Appl. Comput. Harmon. Anal. (to appear)

  5. Boyd, J.P.: Prolate spheroidal wave functions as an alternative to Chebyshev and Legendre polynomials for spectral element and pseudo-spectral algorithms. J. Comput. Phys. 199, 688–716 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  6. Boyd, J.P.: Approximation of an analytic function on a finite real interval by a bandlimited function and conjectures on properties of prolate spheroidal functions. Appl. Comput. Harmon. Anal. 25(2), 168–176 (2003)

    Article  Google Scholar 

  7. Dickinson, R.E.: On exact and approximate linear theory of vertically propagating planetary Rossby waves forced at a spherical lower boundary. Mon. Weather Rev. 96(7), 405–415 (1968)

    Article  Google Scholar 

  8. Dunster, Y.M.: Uniform asymptotic expansions for prolate spheroidal functions with large parameters. SIAM J. Math. Anal. 17(6), 1495–1524 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  9. Flammer, C.: Spheroidal Wave Functions. Stanford University Press, CA (1957)

    MATH  Google Scholar 

  10. Gatteschi, L.: Limitazione degli errori nelle formule asintotiche per le funzioni speciali. (Italian) Univ. e Politec. Torino. Rend. Sem. Mat. 16, 83–94 (1957)

    MathSciNet  Google Scholar 

  11. Glasser, M.L., Klamkin, M.S.: Some integrals of squares of Bessel functions. Util. Math. 12, 315–316 (1977)

    MathSciNet  Google Scholar 

  12. Gosse, L.: Compressed sensing with preconditioning for sparse recovery with subsampled matrices of Slepian prolate functions. Ann. Univ. Ferrara Sez. VII Sci. Math. 59, 81–116 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  13. Hogan, J.A., Lakey, J.D.: Duration and Bandwidth Limiting: Prolate Functions, Sampling, and Applications. Applied and Numerical Harmonic Analysis Series, Birkhäser. Springer, New York (2013)

    Google Scholar 

  14. Karoui, A., Moumni, T.: New efficient methods of computing the prolate spheroidal wave functions and their corresponding eigenvalues. Appl. Comput. Harmon. Anal. 24(3), 269–289 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  15. Landau, H.J., Pollak, H.O.: Prolate spheroidal wave functions, Fourier analysis and uncertainty—III. The dimension of space of essentially time- and band-limited signals. Bell Syst. Tech. J. 41, 1295–1336 (1962)

    Article  MATH  MathSciNet  Google Scholar 

  16. Landau, H.J., Widom, H.: Eigenvalue distribution of time and frequency limiting. J. Math. Anal. Appl. 77, 469–481 (1980)

    Article  MATH  MathSciNet  Google Scholar 

  17. Li, L.W., Kang, X.K., Leong, M.S.: Spheroidal Wave Functions in Electromagnetic Theory. Wiley, London (2001)

    Book  Google Scholar 

  18. Lin, W., Kovvali, N., Carin, L.: Pseudospectral method based on prolate spheroidal wave functions for semiconductor nanodevice simulation. Comput. Phys. Commun. 175, 78–85 (2006)

    Article  MATH  Google Scholar 

  19. Miles, J.W.: Asymptotic approximations for prolate spheroidal wave functions. Stud. Appl. Math. 54(4), 315–349 (1975)

    Article  MATH  MathSciNet  Google Scholar 

  20. Miles, J.W.: Asymptotic eigensolutions of Laplace’s tidal equation. Proc. R. Soc. Lond. A 353(1674), 377–400 (1977)

    Article  MATH  Google Scholar 

  21. Müller, R.: On the structure of the global linearized primitive equations part II: Laplace’s tidal equations. Beitr. Phys. Atmos. 62(2), 112–125 (1989)

    MATH  Google Scholar 

  22. Müller, R.: Stable and unstable eigensolutions of Laplace’s tidal equations for zonal wavenumber zero. Adv. Atmos. Sci. 10(1), 21–40 (1993)

    Article  Google Scholar 

  23. Nikoforov, A.N., Uvarov, V.B.: Special Functions of Mathematical Physics, translated from the Russian edition. Birkhäser, Basel (1988)

  24. Niven, C.: On the conduction of heat in ellipsoids of revolution. Philos. Trans. R. Soc. Lond. 171, 117–151 (1880)

    Article  MATH  Google Scholar 

  25. Oconnor, W.P.: On the application of the spheroidal wave equation to the dynamical theory of the long-period zonal tides in a global ocean. Proc. R. Soc. Lond. A 439(1905), 189–196 (1992)

    Article  Google Scholar 

  26. Olver, F.W.J.: Asymptotics and Special Functions. Academic Press, New York (1974)

    Google Scholar 

  27. Osipov, A.: Certain inequalities involving prolate spheroidal wave functions and associated quantities. Appl. Comput. Harmon. Anal. 35, 359–393 (2013)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  29. Slepian, D.: Prolate spheroidal wave functions, Fourier analysis and uncertainty—IV: extensions to many dimensions; generalized prolate spheroidal functions. Bell Syst. Tech. J. 43, 3009–3057 (1964)

    Article  MATH  MathSciNet  Google Scholar 

  30. Slepian, D.: Some asymptotic expansions for prolate spheroidal wave functions. J. Math. Phys. 44(2), 99–140 (1965)

    Article  MATH  MathSciNet  Google Scholar 

  31. Szegö, G.: Orthogonal Polynomials, 4th edn. American Mathematical Society, Colloquium Publications, vol. XXIII. American Mathematical Society, Providence, RI (1975)

  32. Xiao, H., Rokhlin, V., Yarvin, N.: Prolate spheroidal wave functions, quadrature and interpolation. Inverse Probl. 17, 805–838 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  33. Walter, G., Soleski, T.: A new friendly method of computing prolate spheroidal wave functions and wavelets. Appl. Comput. Harmon. Anal. 19, 432–443 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  34. Wang, L.L.: Analysis of spectral approximations using prolate spheroidal wave functions. Math. Comput. 79(270), 807–827 (2010)

    Article  MATH  Google Scholar 

  35. Watson, G.N.: A Treatise on the Theory of Bessel Functions, 2nd edn. Cambridge University Press, London (1966)

    MATH  Google Scholar 

  36. Widom, H.: Asymptotic behavior of the eigenvalues of certain integral equations. II. Arch. Ration. Mech. Anal. 17, 215–229 (1964)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Aline Bonami.

Additional information

Communicated by Erik Koelink.

This work was supported in part by the ANR Grant “AHPI” ANR-07-BLAN-0247-01, the French–Tunisian CMCU 10G 1503 Project and the DGRST research Grant UR 13 ES 47. Part of this work was done while the second author was visiting the research laboratory MAPMO of the University of Orléans, France.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Bonami, A., Karoui, A. Uniform Approximation and Explicit Estimates for the Prolate Spheroidal Wave Functions. Constr Approx 43, 15–45 (2016). https://doi.org/10.1007/s00365-015-9295-1

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00365-015-9295-1

Keywords

Mathematics Subject Classification

Navigation