Relative asymptotics for orthogonal matrix polynomials

In this paper we study sequences of matrix polynomials that satisfy a non-symmetric recurrence relation. To study this kind of sequences we use a vector interpretation of the matrix orthogonality. In the context of these sequences of matrix polynomials we introduce the concept of the generalized matrix Nevai class and we give the ratio asymptotics between two consecutive polynomials belonging to this class. We study the generalized matrix Chebyshev polynomials and we deduce its explicit expression as well as we show some illustrative examples. The concept of a Dirac delta functional is introduced. We show how the vector model that includes a Dirac delta functional is a representation of a discrete Sobolev inner product. It also allows to reinterpret such perturbations in the usual matrix Nevai class. Finally, the relative asymptotics between a polynomial in the generalized matrix Nevai class and a polynomial that is orthogonal to a modification of the corresponding matrix measure by the addition of a Dirac delta functional is deduced

Asymptotics and zeros of Sobolev orthogonal polynomials on unbounded supports

In this paper we present a survey about analytic properties of polynomials orthogonal with respect to a weighted Sobolev inner product such that the vector of measures has an unbounded support. In particular, we are focused in the study of the asymptotic behaviour of such polynomials as well as in the distribution of their zeros. Some open problems as well as some new directions for a future research are formulated.Comment: Changed content; 34 pages, 41 reference

WKB Approximation and Krall-Type Orthogonal Polynomials

We give a unified approach to the Krall-type polynomials orthogonal withrespect to a positive measure consisting of an absolutely continuous one‘perturbed’ by the addition of one or more Dirac deltafunctions. Some examples studied by different authors are considered from aunique point of view. Also some properties of the Krall-type polynomials arestudied. The three-term recurrence relation is calculated explicitly, aswell as some asymptotic formulas. With special emphasis will be consideredthe second order differential equations that such polynomials satisfy. Theyallow us to obtain the central moments and the WKB approximation of thedistribution of zeros. Some examples coming from quadratic polynomialmappings and tridiagonal periodic matrices are also studied

Coherent pairs of linear functionals on the unit circle

In this paper we extend the concept of coherent pairs of measures from the real line to Jordan arcs and curves. We present a characterization of pairs of coherent measures on the unit circle. Some examples are given

Asymptotic behaviour of Sobolev-type orthogonal polynomials on a rectifiable Jordan arc

22 pages, no figures.-- MSC2000 codes: Primary 42C05.MR#: MR1890494 (2002m:42023)Zbl#: Zbl 0991.42018Our object of study is the asymptotic behavior of the sequence of polynomials orthogonal with respect to the discrete Sobolev inner product $$\langle f, g \rangle = \int_{E} f(\xi) \overline{g(\xi)} \rho (\xi) \xi \xi f(Z) A g(Z)^H,$$ where $E$ is a rectifiable Jordan curve or arc in the complex plane $$f(Z) = (f(z_1), \ldots, f^{(l_1)}(z_1) , \ldots , f(z_m) , \ldots ,f^{(l_m)}(z_m)),$$ $A$ is an $M \times M$ Hermitian matrix, $M=l_{1} + \cdots + l_{m} + m$, $denotes the arc length measure,$\rho$is a nonnegative function on$E$, and$z_{i} \in \Omega$,$i=1,2,\ldots,m$, where$\Omega$is the exterior region to$E$.The work of the first author was supported by the Portuguese Ministry of Science and Technology, Fundação para a Ciência e Tecnología of Portugal under grant FMRH-BSAB-109-99 and by the Centro de Matemática da Universidade de Coimbra. The second author would also like to thank the Unidade de Investigação (Matemática e Aplicações) of the University of Aveiro for their support. The work of the second and third authors was supported by the Dirección General de Enseñanza Superior (DGES) of Spain under grant PB 96-0120-C03-01.Publicad