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Journal of Fourier Analysis and Applications

, Volume 24, Issue 6, pp 1554–1578 | Cite as

The Finite Hankel Transform Operator: Some Explicit and Local Estimates of the Eigenfunctions and Eigenvalues Decay Rates

  • Mourad Boulsane
  • Abderrazek Karoui
Article
  • 68 Downloads

Abstract

For fixed real numbers \(c>0,\)\(\alpha >-\frac{1}{2},\) the finite Hankel transform operator, denoted by \(\mathcal {H}_c^{\alpha }\) is given by the integral operator defined on \(L^2(0,1)\) with kernel \(K_{\alpha }(x,y)= \sqrt{c xy} J_{\alpha }(cxy).\) To the operator \(\mathcal {H}_c^{\alpha },\) we associate a positive, self-adjoint compact integral operator \(\mathcal Q_c^{\alpha }=c\, \mathcal {H}_c^{\alpha }\, \mathcal {H}_c^{\alpha }.\) Note that the integral operators \(\mathcal {H}_c^{\alpha }\) and \(\mathcal Q_c^{\alpha }\) commute with a Sturm-Liouville differential operator \(\mathcal D_c^{\alpha }.\) In this paper, we first give some useful estimates and bounds of the eigenfunctions \(\varphi ^{(\alpha )}_{n,c}\) of \(\mathcal H_c^{\alpha }\) or \(\mathcal Q_c^{\alpha }.\) These estimates and bounds are obtained by using some special techniques from the theory of Sturm-Liouville operators, that we apply to the differential operator \(\mathcal D_c^{\alpha }.\) If \((\mu _{n,\alpha }(c))_n\) and \(\lambda _{n,\alpha }(c)=c\, |\mu _{n,\alpha }(c)|^2\) denote the infinite and countable sequence of the eigenvalues of the operators \(\mathcal {H}_c^{(\alpha )}\) and \(\mathcal Q_c^{\alpha },\) arranged in the decreasing order of their magnitude, then we show an unexpected result that for a given integer \(n\ge 0,\)\(\lambda _{n,\alpha }(c)\) is decreasing with respect to the parameter \(\alpha .\) As a consequence, we show that for \(\alpha \ge \frac{1}{2},\) the \(\lambda _{n,\alpha }(c)\) and the \(\mu _{n,\alpha }(c)\) have a super-exponential decay rate. Also, we give a lower decay rate of these eigenvalues. As it will be seen, the previous results are essential tools for the analysis of a spectral approximation scheme based on the eigenfunctions of the finite Hankel transform operator. Some numerical examples will be provided to illustrate the results of this work.

Keywords

Finite Hankel transform operator Sturm-Liouville operator Eigenfunctions and eigenvalues Prolate spheroidal wave functions Approximation of Hankel band-limited functions 

Mathematics Subject Classification

Primary 42C10 65L70 Secondary 41A60 65L15 

Notes

Acknowledgements

We thank very much the anonymous referees for the valuable comments that helped us to prepare the final version of this work. In particular, we are grateful to the referees for bringing to our attention the reference [17], as well as the other references [18, 20, 21, 23]. This work was supported by the DGRST research Grant UR13ES47 and the Project CMCU PHC Utique 15G1504.

References

  1. 1.
    Abreu, L.D., Bandeira, A.S.: Landau’s necessary conditions for the Hankel transform. J. Funct. Anal. 262(4), 1845–1866 (2012)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Amodio, P., Levitina, T., Settanni, G., Weinmüller, E.B.: On the calculation of the finite Hankel transform eigenfunctions. J. Appl. Math. Comput. 43(1), 151–173 (2013)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Amrein, W.O., Hinz, A.M., Pearson, D.B.: Sturm-Liouville Theory: Past and Present. Birkhäuser, Basel (2005)zbMATHGoogle Scholar
  4. 4.
    Andrews, G.E., Askey, R., Roy, R.: Special Functions. Cambridge University Press, Cambridge, NY (1999)zbMATHGoogle Scholar
  5. 5.
    Bonami, A., Karoui, A.: Uniform bounds of prolate spheroidal wave functions and eigenvalues decay. C. R. Math. Acad. Sci. Paris. Ser. I(352), 229–234 (2014)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Bonami, A., Karoui, A.: Uniform approximation and explicit estimates of the prolate spheroidal wave functions. Constr. Approx. 43(1), 15–45 (2016)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Bonami, A., Karoui, A.: Spectral decay of time and frequency limiting operator. Appl. Comput. Harmon. Anal. 42(1), 1–20 (2017)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Bonami, A., Karoui, A.: Approximations in Sobolev spaces by prolate spheroidal wave functions. Appl. Comput. Harmon. Anal. 42(3), 361–377 (2017)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Breen, S.: Uniform upper and lower bounds on the zeros of Bessel functions of the first kind. J. Math. Anal. Appl. 196, 1–17 (1995)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Elbert, A.: Some recent results on the zeros of Bessel functions and orthogonal polynomials. J. Comput. Appl. Math. 133, 65–83 (2001)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Griffith, J.L.: Hankel transforms of functions zero outside a finite interval. J. Proc. R. Soc. N. S. W. 89, 109–115 (1955)MathSciNetGoogle Scholar
  12. 12.
    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, London (2013)Google Scholar
  13. 13.
    Jaming, P., Karoui, A., Spektor, S.: The approximation of almost time- and band-limited functions by their expansion in some orthogonal polynomials bases. J. Approx. Theory 212, 41–65 (2016)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Karoui, A., Mehrzi, I.: Asymptotic behaviors and numerical computations of the eigenfunctions and eigenvalues associated with the classical and circular prolate spheroidal wave functions. Appl. Math. Comput. 218(22), 10871–10888 (2012)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Karoui, A., Moumni, T.: Spectral analysis of the finite Hankel transform and circular prolate spheroidal wave functions. J. Comput. Appl. Math. 233(2), 315–333 (2009)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Karoui, A., Souabni, A.: Generalized prolate spheroidal wave functions: spectral analysis and approximation of almost band-limited functions. J. Fourier Anal. Appl. 22(2), 383–412 (2016)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Kuznetsov, N.V.: On eigen-functions of an integral equation. In: Zapiski Nauchnykh Seminarov POMI, Mathematical Problems in the Theory of Wave Propagation. Part 3, vol. 17, pp. 66–150 (1970)Google Scholar
  18. 18.
    Landa, B., Shkolnisky, Y.: Approximation scheme for essentially bandlimited and space-concentrated functions on a disk. Appl. Comput. Harmon. Anal. 43(3), 381–403 (2017)MathSciNetzbMATHGoogle Scholar
  19. 19.
    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)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Landau, H.J., Widom, H.: Eigenvalue distribution of time and frequency limiting. J. Math. Anal. Appl. 77, 469–481 (1980)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Levitina, T.: On the eigenfunctions of the finite Hankel transform. Sampl. Theory Signal Image Process. 11(1), 55–79 (2012)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Osipov, A., Rokhlin, V., Xiao, H.: Prolate Spheroidal Wave Functions of Order Zero, Applied Mathematical Sciences, vol. 187. Springer, New York (2013)zbMATHGoogle Scholar
  23. 23.
    Shkolnisky, Y.: Prolate spheroidal wave functions on a discIntegration and approximation of two-dimensional bandlimited functions. Appl. Comput. Harmon. Anal. 22(2), 235–256 (2007)MathSciNetzbMATHGoogle Scholar
  24. 24.
    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)zbMATHGoogle Scholar
  25. 25.
    Slepian, D.: Some asymptotic expansions for prolate spheroidal wave functions. J. Math. Phys. 44(2), 99–140 (1965)MathSciNetzbMATHGoogle Scholar
  26. 26.
    Watson, G.N.: A Treatise on the Theory of Bessel Functions, 2nd edn. Cambridge University Press, London (1966)zbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of Sciences of BizerteUniversity of CarthageJarzounaTunisia

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