Operator theoretical treatments of \(\epsilon\)-entropy of a Gaussian process per unit time (Q1103962)

The purpose of this paper is to study the \(\epsilon\)-entropy of a convolution operator per unit time on a functional Hilbert space \(L^ 2[0,T]\) with \(T>0\). A relation between the proper values of a convolution operator and the spectral density of its kernel function is discussed. Namely, we generalize the result of \textit{M. Kac}, \textit{W. L. Murdock} and \textit{G. Szegö} [J. Ration. Mech. Anal. 2, 767-800 (1953; Zbl 0051.303)]. This generalization is applied to the characterization of \(\epsilon\)-entropy of a Gaussian process per unit time by the sequence of \(\epsilon\)-entropies of integral kernel operators per unit time, which are induced by the covariance function of this process.