### Spectral transformations of hermitian linear functionals

In this talk we present some recent results concerning rational modications of moment functionals on the

unit circle, i.e., associated with hermitian Toeplitz matrices. We present a new approach which allows us to

study arbitrary rational modications. The main objective is the characterization of the quasi-deniteness

of the functionals involved in the problem in terms of a dierence equation relating the corresponding Schur

parameters. The results are presented in the general framework of (non necessarily quasi-denite) 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.

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68 CHAPTER 7. FRIDAY, SEPTEMBER 2

The eectiveness of this new approach is shown in several examples which also lead to the discovery

of new families of orthogonal polynomials on the unit circle, some of them with respect to non positive

denite functionals.

Paper joint with L. Moral and L. Velázquez

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