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 KarouiEmail author


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.


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 



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.


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© 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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