The finite Hankel transform operator: Some explicit and local estimates of the eigenfunctions and eigenvalues decay rates

摘要

For fixed real numbers $c>0,$ $\alpha>-\frac{1}{2},$ the finite Hankeltransform operator, denoted by $\mathcal{H}_c^{\alpha}$ is given by theintegral operator defined on $L^2(0,1)$ with kernel $K_{\alpha}(x,y)= \sqrt{cxy} J_{\alpha}(cxy).$ To the operator $\mathcal{H}_c^{\alpha},$ we associate apositive, self-adjoint compact integral operator $\mathcal Q_c^{\alpha}=c\,\mathcal{H}_c^{\alpha}\, \mathcal{H}_c^{\alpha}.$ Note that the integraloperators $\mathcal{H}_c^{\alpha}$ and $\mathcal Q_c^{\alpha}$ commute with aSturm-Liouville differential operator $\mathcal D_c^{\alpha}.$ In this paper, we first give some useful estimates and bounds of theeigenfunctions $\vp$ of $\mathcal H_c^{\alpha}$ or $\mathcal Q_c^{\alpha}.$These estimates and bounds are obtained by using some special techniques fromthe theory of Sturm-Liouville operators, that we apply to the differentialoperator $\mathcal D_c^{\alpha}.$ 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 thedecreasing order of their magnitude, then we show an unexpected result that fora given integer $n\geq 0,$ $\lambda_{n,\alpha}(c)$ is decreasing with respectto the parameter $\alpha.$ As a consequence, we show that for $\alpha\geq\frac{1}{2},$ the $\lambda_{n,\alpha}(c)$ and the $\mu_{n,\alpha}(c)$ have asuper-exponential decay rate. Also, we give a lower decay rate of theseeigenvalues. As it will be seen, the previous results are essential tools forthe analysis of a spectral approximation scheme based on the eigenfunctions ofthe finite Hankel transform operator. Some numerical examples will be providedto illustrate the results of this work.