Operator Bezoutiant and roots of entire functions, concrete examples

Abstract

The matrix Bezoutiant is used in order to define the number of common zeroes of two polynomials $f(z)$ and $g(z)$and to describe the distribution of the zeroes of polynomials with respect to the circle $|z|=1$ (see [81]). M.G. Krein extended the notion of Bezoutiant to entire functions of the form$F(z) = 1+\int^a_0 e^{izt}\overline{\Phi(t)}dt,\,\,\,\, \Phi(t)\,\epsilon \,L(0,a).$