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, $O (e^{-(\pi-h\sigma)N/2}/\sqrt{N} )$ , 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 $O (e^{-(2\pi-h\sigma)N/2}/\sqrt {N} )$ . Numerical examples are given with comparisons with the best sampling technique up to now, i.e. sinc-Gaussian.
Highlights
Let Eσ (φ), σ >, be the class of entire functions satisfying the following condition:f (ζ ) ≤ φ | ζ | eσ| ζ|, ζ ∈ C, ( . )where φ is a non-decreasing, non-negative function on [, ∞)
This paper is concerned with constructing a new sampling technique to approximate eigenvalues of Sturm-Liouville problems with separate-type boundary conditions using Hermite-Gauss operator Hh,N
In this paper we need the bound of amplitude error only on a real domain because the eigenvalues of Sturm-Liouville problem ( . )-( . ) are real numbers but in the general cases the eigenvalues are not necessarily real and this technique will be used for approximating eigenvalues of different classes of boundary value problems
Summary
Let Eσ (φ), σ > , be the class of entire functions satisfying the following condition:f (ζ ) ≤ φ | ζ | eσ| ζ|, ζ ∈ C, ( . )where φ is a non-decreasing, non-negative function on [ , ∞). The authors of [ ] investigated a bound of the approximating function from the class Eσ (φ) by the sinc-Gaussian operator. Annaby and Asharabi [ ] have constructed a new sampling technique to approximate eigenvalues of second order Birkhoff-regular eigenvalue problems using sinc-Gaussian operator.
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