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

Abstract 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 − h σ ) N / 2 / N ) $O (e^{-(\pi-h\sigma)N/2}/\sqrt{N} )$ , 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 − ( 2 π − h σ ) N / 2 / N ) $O (e^{-(2\pi-h\sigma)N/2}/\sqrt {N} )$ . Numerical examples are given with comparisons with the best sampling technique up to now, i.e. sinc-Gaussian