Resumen de Spectral problems and orthogonal polynomials on the unit circle
Kenier Castillo 
The main purpose of the work presented here is to study transformations of sequences of orthogonal polynomials associated with a hermitian linear functional L, using spectral transformations of the corresponding C-function F. We show that a rational spectral transformation of F is given by a finite composition of four canonical spectral transformations. In addition to the canonical spectral transformations, we deal with two new examples of linear spectral transformations. First, we analyze a spectral transformation of L such that the corresponding moment matrix is the result of the addition of a constant on the main diagonal or on two symmetric sub-diagonals of the initial moment matrix. Next, we introduce a spectral transformation of L by the addition of the first derivative of a complex Dirac linear functional when its support is a point on the unit circle or two points symmetric with respect to the unit circle. In this case, outer relative asymptotics for the new sequences of orthogonal polynomials in terms of the original ones are obtained. Necessary and su cient conditions for the quasi-definiteness of the new linear functionals are given. The relation between the corresponding sequence of orthogonal polynomials in terms of the original one is presented. We also consider polynomials which satisfy the same recurrence relation as the polynomials orthogonal with respect to the linear functional L , with the restriction that the Verblunsky coe cients are in modulus greater than one. With positive or alternating positive-negative values for Verblunsky coe cients, zeros, quadrature rules, integral representation, and associated moment problem are analyzed. We also investigate the location, monotonicity, and asymptotics of the zeros of polynomials orthogonal with respect to a discrete Sobolev inner product for measures supported on the real line and on the unit circle. -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
