Spectral transformations of hermitian linear functionals
In this talk we present some recent results concerning rational modi cations 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 modi cations. The main objective is the characterization of the quasi-de niteness
of the functionals involved in the problem in terms of a di erence equation relating the corresponding Schur
parameters. The results are presented in the general framework of (non necessarily quasi-de nite) 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 e ectiveness 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
de nite functionals.
Paper joint with L. Moral and L. Velázquez
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