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- M J Cantero, L Moral, L Velázquez
- 2005

In this paper a general theory of semi-classical matrix orthogonal polynomials is developed. We define the semi-classical linear functionals by means of a distri-butional equation D(uA) = uB, where A and B are matrix polynomials. Several characterizations for these semi-classical functionals are given in terms of the corresponding (left) matrix orthogonal… (More)

- L. Velázquez
- 2004

In this paper we prove that the simplest band representations of unitary operators on a Hilbert space are five-diagonal. Orthogonal polynomials on the unit circle play an essential role in the development of this result, and also provide a parametrization of such five-diagonal representations which shows specially simple and interesting decomposition and… (More)

In this paper we obtain new results about the orthogonality measure of orthogonal polynomials on the unit circle, through the study of unitary truncations of the corresponding unitary multiplication operator , and the use of the five-diagonal representation of this operator. Unitary truncations on subspaces with finite co-dimension give information about… (More)

- L. Velázquez
- 2008

An operator theoretic approach to orthogonal rational functions on the unit circle with poles in its exterior is presented in this paper. This approach is based on the identification of a suitable matrix representation of the multiplication operator associated with the corresponding orthogonality measure. Two different alternatives are discussed, depending… (More)

- M J Cantero, M P Ferrer, L Moral, L Velázquez
- 2002

Szeg˝ o's procedure to connect orthogonal polynomials on the unit circle and orthogonal polynomials on [−1, 1] is generalized to nonsymmetric measures. It generates the so-called semi-orthogonal functions on the linear space of Laurent polynomials Λ, and leads to a new orthogonality structure in the module Λ × Λ. This structure can be interpreted in terms… (More)

In this paper we prove some characterizations of the matrix orthogonal polynomials whose derivatives are also orthogonal, which generalize other known ones in the scalar case. In particular, we prove that the corresponding orthogonality matrix functional is characterized by a Pearson-type equation with two matrix polynomials of degree not greater than 2 and… (More)

- M. J. Cantero, F. Marcellán, L. Moral, L. Velázquez
- 2015

We develop a theory of Darboux transformations for CMV matrices, canonical representations of the unitary operators. In perfect analogy with their self-adjoint version – the Darboux transformations of Jacobi matrices – they are equivalent to Laurent polynomial modifications of the underlying measures. We address other questions which emphasize the… (More)

- L. Velazquez
- 2008

Microcanonical description is characterized by the presence of an internal symmetry closely related with the dynamical origin of this ensemble: the reparametrization invariance. Such symmetry possibilities the development of a non Riemannian geometric formulation within the microcanonical description of an isolated system, which leads to an unexpected… (More)