A Hermite-Gauss method for the approximation of eigenvalues of regular Sturm-Liouville problems

Recently, some authors have used the sinc-Gaussian sampling technique to approximate eigenvalues of boundary value problems rather than the classical sinc technique because the sinc-Gaussian technique has a convergence rate of the exponential order, O(e−(π−hσ)N/2/N)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$O (e^{-(\pi-h\sigma)N/2}/\sqrt{N} )$\end{document}, where σ, h are positive numbers and N is the number of terms in sinc-Gaussian technique. As is well known, the other sampling techniques (classical sinc, generalized sinc, Hermite) have a convergence rate of a polynomial order. In this paper, we use the Hermite-Gauss operator, which is established by Asharabi and Prestin (Numer. Funct. Anal. Optim. 36:419-437, 2015), to construct a new sampling technique to approximate eigenvalues of regular Sturm-Liouville problems. This technique will be new and its accuracy is higher than the sinc-Gaussian because Hermite-Gauss has a convergence rate of order O(e−(2π−hσ)N/2/N)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$O (e^{-(2\pi-h\sigma)N/2}/\sqrt {N} )$\end{document}. Numerical examples are given with comparisons with the best sampling technique up to now, i.e. sinc-Gaussian.