The square-transform of hermite-biehler functions. A geometric approach

Abstract

We investigate the subclass of symmetric indefinite Hermite-Biehler functions which is obtained from positive definite Hermite-Biehler functions by means of the square-transform. It is known that functions of this class can be characterized in terms of location of their zeros. We give another, more elementary and geometric, proof of this result. The present proof employs a ‘shifting-of-zeros’ perturbation method. We apply our results to obtain information on the eigenvalues of a concrete boundary value problems.