On Numerical Computation of the Spectrum of a Class of Convolution Operators Related to Delay Systems

Abstract

A class of convolution operators on fixed interval with initial values is considered. The numerical computation of its spectra is attempted via a finite-dimensional approximation so-called fast sampling and hold. In contrast to the case of matrices, the spectrum of an operator is not continuous against small perturbations in general. This implies that a fine approximation does not necessarily lead to better estimates close to the true values. Therefore, one must provide a mathematical justification of the procedure depending on the operator of interest. In this paper, continuity of the spectrum of the convolution operator described above is proved and a numerical computation formula for the calculation of its eigenvalues is given.