be the Paley-Wiener space of all
-entire functions of exponential type σ
. Assume that
can be reconstructed via the Hermite-type sampling series
is the sequences of sinc functions
Series (1.1) converges absolutely and uniformly on ℝ, cf.
]. Sometimes, series (1.1) is called the derivative sampling theorem. Our task is to use formula (1.1) to compute eigenvalues of Dirac systems numerically. This approach is a fully new technique that uses the recently obtained estimates for the truncation and amplitude errors associated with (1.1), cf.
]. Both types of errors normally appear in numerical techniques that use interpolation procedures. In the following we summarize these estimates. The truncation error associated with (1.1) is defined to be
is the truncated series
It is proved in [5
] that if
is sufficiently smooth in the sense that there exists
, then, for
, we have
where the constants
are given by
The amplitude error occurs when approximate samples are used instead of the exact ones, which we cannot compute. It is defined to be
are approximate samples of
, respectively. Let us assume that the differences
, are bounded by a positive number ε
satisfies the natural decay conditions
, then for
, we have, [5
and is the Euler-Mascheroni constant.
The classical [6
] sampling theorem of Whittaker, Kotel’nikov and Shannon (WKS) for
is the series representation
where the convergence is absolute and uniform on ℝ and it is uniform on compact sets of ℂ, cf.
]. Series (1.12), which is of Lagrange interpolation type, has been used to compute eigenvalues of second-order eigenvalue problems; see, e.g.
]. The use of (1.12) in numerical analysis is known as the sinc-method established by Stenger, cf.
]. In [10
], the authors applied (1.12) and the regularized sinc-method to compute eigenvalues of Dirac systems with a derivation of the error estimates as given by [19
]. In [12
] the Dirac system has an eigenparameter appearing in the boundary conditions. The aim of this paper is to investigate the possibilities of using Hermite interpolations rather than Lagrange interpolations, to compute the eigenvalues numerically. Notice that, due to Paley-Wiener’s theorem [21
if and only if there is
also has an expansion of the form (1.12). However,
can be also obtained by the term-by-term differentiation formula of (1.12)
see [, p.52] for convergence. Thus the use of Hermite interpolations will not cost any additional computational efforts since the samples will be used to compute both and according to (1.12) and (1.14), respectively.
Consider the Dirac system which consists of the system of differential equations
and the boundary conditions
The eigenvalue problem (1.15)-(1.17) will be denoted by when . It is a Dirac system when the eigenparameter λ appears linearly in both boundary conditions. The classical problem when , which we denote by , is studied in the monographs of Levitan and Sargsjan [22, 23]. Annaby and Tharwat  used Hermite-type sampling series (1.1) to compute the eigenvalues of problem numerically. In , Kerimov proved that has a denumerable set of real and simple eigenvalues with ±∞ as the limit points. Similar results are established in  for the problem when the eigenparameter appears in one condition, i.e., when , or equivalently when and , where also sampling theorems have been established. These problems will be denoted by and , respectively. The aim of the present work is to compute the eigenvalues of and numerically by the Hermite interpolations with an error analysis. This method is based on sampling theorem, Hermite interpolations, but applied to regularized functions hence avoiding any (multiple) integration and keeping the number of terms in the Cardinal series manageable. It has been demonstrated that the method is capable of delivering higher-order estimates of the eigenvalues at a very low cost; see . In Sections 2 and 3, we derive the Hermite interpolation technique to compute the eigenvalues of Dirac systems with error estimates. We briefly derive some necessary asymptotics for Dirac systems’ spectral quantities. The last section contains three worked examples with comparisons accompanied by figures and numerics with the Lagrange interpolation method.