This paper is devoted to the study of general (Laurent) polynomial
modifications of moment functionals on the unit circle, i.e., associated with
hermitian Toeplitz matrices. We present a new approach which allows us to study
polynomial modifications of arbitrary degree.

The main objective is the characterization of the quasi-definiteness of the
functionals involved in the problem in terms of a difference equation relating
the corresponding Schur parameters. The results are presented in the general
framework of (non necessarily quasi-definite) hermitian functionals, so that
the maximum number of orthogonal polynomials is characterized by the number of
consistent steps of an algorithm based on the referred recurrence for the Schur
parameters.

Some concrete applications to the study of orthogonal polynomials on the unit
circle show the effectiveness of this new approach: an exhaustive and
instructive analysis of the functionals coming from a general inverse
polynomial perturbation of degree one for the Lebesgue measure; the
classification of those pairs of orthogonal polynomials connected by a kind of
linear relation with constant polynomial coefficients; and the determination of
those orthogonal polynomials whose associated ones are related to a degree one
polynomial modification of the original orthogonality functional.