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This paper is devoted to the study of general (Laurent) polynomialmodifications of moment functionals on the unit circle, i.e., associated withhermitian Toeplitz matrices. We present a new approach which allows us to studypolynomial modifications of arbitrary degree.  The main objective is the characterization of the quasi-definiteness of thefunctionals involved in the problem in terms of a difference equation relatingthe corresponding Schur parameters. The results are presented in the generalframework of (non necessarily quasi-definite) hermitian functionals, so thatthe maximum number of orthogonal polynomials is characterized by the number ofconsistent steps of an algorithm based on the referred recurrence for the Schurparameters.  Some concrete applications to the study of orthogonal polynomials on the unitcircle show the effectiveness of this new approach: an exhaustive andinstructive analysis of the functionals coming from a general inversepolynomial perturbation of degree one for the Lebesgue measure; theclassification of those pairs of orthogonal polynomials connected by a kind oflinear relation with constant polynomial coefficients; and the determination ofthose orthogonal polynomials whose associated ones are related to a degree onepolynomial modification of the original orthogonality functional.

Polynomial perturbations of hermitian linear functionals and difference equations