María José Cantero

Five-diagonal matrices and zeros of orthogonal polynomials on the unit circle

It is shown that monic orthogonal polynomials on the unit circle are the characteristic polynomials of certain five-diagonal matrices depending on the Schur parameters. This result is achieved through the study of orthogonal Laurent polynomials on the unit circle. More precisely, it is a consequence of the five term recurrence relation obtained for these orthogonal Laurent polynomials, and the one to one correspondence established between them and the orthogonal polynomials on the unit circle. As an application, some results relating the behaviour of the zeros of orthogonal polynomials and the location of Schur parameters are obtained.

The CGMV method for quantum walks

We review the main aspects of a recent approach to quantum walks, the CGMV method. This method proceeds by reducing the unitary evolution to canonical form, given by the so-called CMV matrices, which act as a link to the theory of orthogonal polynomials on the unit circle. This connection allows one to obtain results for quantum walks which are hard to tackle with other methods. Behind the above connections lies the discovery of a new quantum dynamical interpretation for well known mathematical tools in complex analysis. Among the standard examples which will illustrate the CGMV method are the famous Hadamard and Grover models, but we will go further showing that CGMV can deal even with non-translation invariant quantum walks. CGMV is not only a useful technique to study quantum walks, but also a method to construct quantum walks à la carte. Following this idea, a few more examples illustrate the versatility of the method. In particular, a quantum walk based on a construction of a measure on the unit circle due to F. Riesz will point out possible non-standard behaviours in quantum walks.

Matrix orthogonal polynomials whose derivatives are also orthogonal

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 1. The proofs are given for a general sequence of matrix orthogonal polynomials, not necessarily associated with a hermitian functional. We give several examples of non-diagonalizable positive definite weight matrices satisfying a Pearson-type equation, which show that the previous results are non-trivial even in the positive definite case. A detailed analysis is made for the class of matrix functionals which satisfy a Pearson-type equation whose polynomial of degree not greater than 2 is scalar. We characterize the Pearson-type equations of this kind that yield a sequence of matrix orthogonal polynomials, and we prove that these matrix orthogonal polynomials satisfy a second order differential equation even in the non-hermitian case. Finally, we prove and improve a conjecture of Duran and Grunbaum concerning the triviality of this class in the positive definite case, while some examples show the non-triviality for hermitian functionals which are not positive definite.

A matricial computation of rational quadrature formulas on the unit circle

A matricial computation of quadrature formulas for orthogonal rational functions on the unit circle, is presented in this paper. The nodes of these quadrature formulas are the zeros of the para-orthogonal rational functions with poles in the exterior of the unit circle and the weights are given by the corresponding Christoffel numbers. We show how these nodes can be obtained as the eigenvalues of the operator Möbius transformations of Hessenberg matrices and also as the eigenvalues of the operator Möbius transformations of five-diagonal matrices, recently obtained. We illustrate the preceding results with some numerical examples.

On Rapid Computation of Expansions in Ultraspherical Polynomials

We present an

Direct and inverse polynomial perturbations of hermitian linear functionals

{\cal O}(N\log_2N)

Poisson brackets of orthogonal polynomials

algorithm for the computation of the first

A Class of Nonsymmetric Orthogonal Polynomials on the Unit Circle

N

Full length article: An extension of the associated rational functions on the unit circle

coefficients in the expansion of an analytic function in ultraspherical polynomials. We first represent expansion coefficients as an infinite linear combination of derivatives and then as an integral transform with a hypergeometric kernel along the boundary of a Bernstein ellipse. Following a transformation of the kernel, we approximate the coefficients to arbitrary accuracy using the discrete Fourier transform.

Semiorthogonal functions and orthogonal polynomials on the unit circle

Abstract This paper is devoted to the study of direct and inverse (Laurent) polynomial modifications of moment functionals on the unit circle, i.e., associated with hermitian Toeplitz matrices. We present a new approach which allows us to study polynomial modifications of arbitrary degree. The main objective is the characterization of the quasi-definiteness of the functionals involved in the problem in terms of a difference equation relating the corresponding Schur parameters. The results are presented in the general framework of (non-necessarily quasi-definite) 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. The non-uniqueness of the inverse problem makes it more interesting than the direct one. Due to this reason, special attention is paid to the inverse modification, showing that different approaches are possible depending on the data about the polynomial modification at hand. These different approaches are translated as different kinds of initial conditions for the related inverse algorithm. Some concrete applications to the study of orthogonal polynomials on the unit circle show the effectiveness of this new approach: an exhaustive and instructive analysis of the functionals coming from a general inverse polynomial perturbation of degree one for the Lebesgue measure; the classification of those pairs of orthogonal polynomials connected by a certain type of linear relation with constant polynomial coefficients; and the determination of those orthogonal polynomials whose associated ones are related to a degree one polynomial modification of the original orthogonality functional.