LAURENT POLYNOMIAL PERTURBATIONS OF LINEAR FUNCTIONALS. AN INVERSE PROBLEM.

Abstract

Given a linear functional L in the linear space P of polynomials with complex coefficients, we analyze those linear functionals (L) over tilde such that, for a fixed alpha is an element of C, <(L) over tilde, (z + z(-1) - (alpha + (alpha) over bar))p > = < L, p > for every p is an element of P. We obtain the relation between the corresponding Caratheodory functions in such a way that a linear spectral transform appears. If L is a positive definite linear functional, the necessary and sufficient conditions in order for (L) over tilde to be a quasi-definite linear functional are given. The relation between the corresponding sequences of monic orthogonal polynomials is presented.