arXiv subjects

Abey López García

Publications and source records attributed to Abey López García.

6 recordsLinked to original sources

arXiv2019

On random polynomials generated by a symmetric three-term recurrence relation

We investigate the sequence $(P_{n}(z))_{n=0}^{\infty}$ of random polynomials generated by the three-term recurrence relation $P_{n+1}(z)=z P_{n}(z)-a_{n} P_{n-1}(z)$, $n\geq 1$, with initial conditions $P_{\ell}(z)=z^{\ell}$, $\ell=0, 1$, assuming that $(a_{n})_{n\in\mathbb{Z}}$ is a sequence of positive i.i.d. random variables. $(P_{n}(z))_{n=0}^{\infty}$ is a sequence of orthogonal polynomials on the real line, and $P_{n}$ is the characteristic polynomial of a Jacobi matrix $J_{n}$. We investigate the relation between the common distribution of the recurrence coefficients $a_{n}$ and two other distributions obtained as weak limits of the averaged empirical and spectral measures of $J_{n}$. Our main result is a description of combinatorial relations between the moments of the aforementioned distributions in terms of certain classes of colored planar trees. Our approach is combinatorial, and the starting point of the analysis is a formula of P. Flajolet for weight polynomials associated with labelled Dyck paths.

math.PR↗

arXiv2012

A family of Nikishin systems with periodic recurrence coefficients

Suppose we have a Nikishin system of $p$ measures with the $k$th generating measure of the Nikishin system supported on an interval $Δ_k\subset\er$ with $Δ_k\capΔ_{k+1}=\emptyset$ for all $k$. It is well known that the corresponding staircase sequence of multiple orthogonal polynomials satisfies a $(p+2)$-term recurrence relation whose recurrence coefficients, under appropriate assumptions on the generating measures, have periodic limits of period $p$. (The limit values depend only on the positions of the intervals $Δ_k$.) Taking these periodic limit values as the coefficients of a new $(p+2)$-term recurrence relation, we construct a canonical sequence of monic polynomials $\{P_{n}\}_{n=0}^{\infty}$, the so-called \emph{Chebyshev-Nikishin polynomials}. We show that the polynomials $P_{n}$ themselves form a sequence of multiple orthogonal polynomials with respect to some Nikishin system of measures, with the $k$th generating measure being absolutely continuous on $Δ_{k}$. In this way we generalize a result of the third author and Rocha \cite{LopRoc} for the case $p=2$. The proof uses the connection with block Toeplitz matrices, and with a certain Riemann surface of genus zero. We also obtain strong asymptotics and an exact Widom-type formula for the second kind functions of the Nikishin system for $\{P_{n}\}_{n=0}^{\infty}$.

math.CA↗

arXiv2012

High order three-term recursions, Riemann-Hilbert minors and Nikishin systems on star-like sets

We study monic polynomials $Q_n(x)$ generated by a high order three-term recursion $xQ_n(x)=Q_{n+1}(x)+a_{n-p} Q_{n-p}(x)$ with arbitrary $p\geq 1$ and $a_n>0$ for all $n$. The recursion is encoded by a two-diagonal Hessenberg operator $H$. One of our main results is that, for periodic coefficients $a_n$ and under certain conditions, the $Q_n$ are multiple orthogonal polynomials with respect to a Nikishin system of orthogonality measures supported on star-like sets in the complex plane. This improves a recent result of Aptekarev-Kalyagin-Saff where a formal connection with Nikishin systems was obtained in the case when $\sum_{n=0}^{\infty}|a_n-a|<\infty$ for some $a>0$. An important tool in this paper is the study of "Riemann-Hilbert minors", or equivalently, the "generalized eigenvalues" of the Hessenberg matrix $H$. We prove interlacing relations for the generalized eigenvalues by using totally positive matrices. In the case of asymptotically periodic coefficients $a_n$, we find weak and ratio asymptotics for the Riemann-Hilbert minors and we obtain a connection with a vector equilibrium problem. We anticipate that in the future, the study of Riemann-Hilbert minors may prove useful for more general classes of multiple orthogonal polynomials.

math.CA↗

arXiv2010

Asymptotics of multiple orthogonal polynomials for a system of two measures supported on a starlike set

For a system of two measures supported on a starlike set in the complex plane, we study asymptotic properties of associated multiple orthogonal polynomials $Q_{n}$ and their recurrence coefficients. These measures are assumed to form a Nikishin-type system, and the polynomials $Q_{n}$ satisfy a three-term recurrence relation of order three with positive coefficients. Under certain assumptions on the orthogonality measures, we prove that the sequence of ratios $\{Q_{n+1}/Q_{n}\}$ has four different periodic limits, and we describe these limits in terms of a conformal representation of a compact Riemann surface. Several relations are found involving these limiting functions and the limiting values of the recurrence coefficients. We also study the $n$th root asymptotic behavior and zero asymptotic distribution of $Q_{n}$.

math.CV↗

arXiv2008

Relative Asymptotic of Multiple Orthogonal Polynomials for Nikishin Systems

We prove relative asymptotic for the ratio of two sequences of multiple orthogonal polynomials with respect to Nikishin system of measures. The first Nikishin system ${\mathcal{N}}(σ_1,...,σ_m)$ is such that for each $k$, $σ_k$ has constant sign on its compact support $\supp {σ_k} \subset \mathbb{R}$ consisting of an interval $\widetildeΔ_k$, on which $|σ_k^{\prime}| > 0$ almost everywhere, and a discrete set without accumulation points in $\mathbb{R} \setminus \widetildeΔ_k$. If ${Co}(\supp {σ_k}) = Δ_k$ denotes the smallest interval containing $\supp {σ_k}$, we assume that $Δ_k \cap Δ_{k+1} = \emptyset$, $k=1,...,m-1$. The second Nikishin system ${\mathcal{N}}(r_1σ_1,...,r_mσ_m)$ is a perturbation of the first by means of rational functions $r_k$, $k=1,...,m,$ whose zeros and poles lie in $\mathbb{C} \setminus \cup_{k=1}^m Δ_k$.

math.CV↗

arXiv2007

Ratio Asymptotic of Hermite-Padé Orthogonal Polynomials for Nikishin Systems. II

We prove ratio asymptotic for sequences of multiple orthogonal polynomials with respect to a Nikishin system of measures ${\mathcal{N}}(σ_1,...,σ_m)$ such that for each $k$, the support of $σ_k$ consists of an interval $\widetildeΔ_k$, on which $σ_k^{\prime} > 0$ almost everywhere, and a set without accumulation points in $\mathbb{R} \setminus \widetildeΔ_k$.

math.CV↗