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

## TL;DR

This paper analyzes the eigenfunctions and eigenvalues of the finite Hankel transform operator, providing bounds, decay rates, and monotonicity properties, with implications for spectral approximation methods.

## Contribution

It offers new estimates for eigenfunctions, reveals monotonicity of eigenvalues with respect to parameter alpha, and establishes super-exponential decay rates for eigenvalues when alpha is at least 0.5.

## Key findings

- Eigenfunctions have specific bounds and estimates derived from Sturm-Liouville theory.
- Eigenvalues decrease with increasing alpha for fixed n and c.
- Eigenvalues exhibit super-exponential decay for alpha ≥ 0.5.

## 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 $\vp$ 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 $n\geq 0,$ $\lambda_{n,\alpha}(c)$ is decreasing with respect to 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 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.