PROPERTIES OF CHARACTERISTIC FREQUENCIES OF LINEAR EQUATIONS OF ARBITRARY ORDER

摘要

New characteristics of oscillation properties of solutions are suggested, namely, the frequencies equal to the upper or the lower time-average of the null-points, sign alteration points, or roots (their multiplicity being taken into account). We examine several definitions of the principal values of frequencies of a linear homogeneous equation; for an equation with constant coefficients, all these values coincide with the moduli of the imaginary parts of the roots of the corresponding characteristic polynomial. It is shown that for equations of an arbitrary (＞ 2) order, these definitions yield different values, which are, in general, unstable with respect to uniformly small, or even infinitesimal, perturbations of coefficients, and the extreme values belong to precisely the second Baire class, both in the uniform and the compact-open topologies.