Nikishin systems are perfect

K. Mahler introduced the concept of perfect systems in the general theory he developed for the simultaneous Hermite-Pade approximation of analytic functions. We prove that Nikishin systems are perfect providing, by far, the largest class of systems of functions for which this important property holds. As consequences, in the context of Nikishin systems, we obtain: an extension of Markov's theorem to simultaneous Hermite-Pade approximation, a general result on the convergence of simultaneous quadrature rules of Gauss-Jacobi type, the logarithmic asymptotics of general sequences of multiple orthogonal polynomials, and an extension of the Denisov-Rakhmanov theorem for the ratio asymptotics of mixed type multiple orthogonal polynomials.Comment: 39 page

Ratio and relative asymptotics of polynomials orthogonal with respect to varying Denisov-type measures

34 pages, no figures.-- MSC1991 codes: 42C05, 41A28.-- Dedicated to Barry Simon on the occasion of his sixtieth birthday.MR#: MR2220040 (2006m:42002)Zbl#: Zbl 1100.42014Let μ be a finite positive Borel measure with compact support consisting of an interval [c,d] ⊂ R plus a set of isolated points in R\[c,d], such that μ′>0 almost everywhere on [c,d]. Let $\{w_{2n}\}$, $n\in\Bbb Z_+$, be a sequence of polynomials, $\deg w_{2n}\leq2n$, with real coefficients whose zeros lie outside the smallest interval containing the support of μ. We prove ratio and relative asymptotics of sequences of orthogonal polynomials with respect to varying measures of the form $\frac{d\mu_n}{w_{2n}}$. In particular, we obtain an analogue for varying measures of Denisov's extension of Rakhmanov's theorem on ratio asymptotics. These results on varying measures are applied to obtain ratio asymptotics for orthogonal polynomials with respect to fixed measures on the unit circle and for multi-orthogonal polynomials in which the measures involved are of the type described above.Research of D.B. Rolanía was partially supported by Dirección General de Investigación, Ministerio de Ciencia y Tecnología, under Grant BFM 2003-06335-C03-02. Research of de la Calle Ysern was supported by Dirección General de Investigación, Ministerio de Ciencia y Tecnología, under Grants BFM 2002-04315-C02-01 and BFM 2003-06335-C03-02. Research of G.L. Lagomasino was supported by Grants INTAS 03-516637, NATO PST.CLG.979738, and by Dirección General de Investigación, Ministerio de Ciencia y Tecnología, under Grant BFM 2003-06335-C03-02.Publicad

Uniform Asymptotics of Orthogonal Polynomials Arising from Coherent States

In this paper, we study a family of orthogonal polynomials $\{\phi_n(z)\}$ arising from nonlinear coherent states in quantum optics. Based on the three-term recurrence relation only, we obtain a uniform asymptotic expansion of $\phi_n(z)$ as the polynomial degree $n$ tends to infinity. Our asymptotic results suggest that the weight function associated with the polynomials has an unusual singularity, which has never appeared for orthogonal polynomials in the Askey scheme. Our main technique is the Wang and Wong's difference equation method. In addition, the limiting zero distribution of the polynomials $\phi_n(z)$ is provided

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

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