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.     

Zeros and logarithmic asymptotics of Sobolev orthogonal polynomials for exponential weights

We obtain the (contracted) weak zero asymptotics for orthogonal polynomials with respect to Sobolev inner products with exponential weights in the real semiaxis, of the form , with γ>0, which include as particular cases the counterparts of the so-called Freud (i.e., when φ has a polynomial growth at infinity) and Erdös (when φ grows faster than any polynomial at infinity) weights. In addition, the boundness of the distance of the zeros of these Sobolev orthogonal polynomials to the convex hull of the support and, as a consequence, a result on logarithmic asymptotics are derived.     

Strong asymptotics for Sobolev orthogonal polynomials

In this paper we obtain the strong asymptotics for the sequence of orthogonal polynomials with respect to the inner product $\left\langle {f,g} \right\rangle s = \sum\limits_{k - 0}^m {\int\limits_{\Delta _k } {f^{\left( k \right)} \left( x \right)g^{\left( k \right)} \left( x \right)d\mu \kappa } } \left( x \right)$ where $\left\{ {\mu _\kappa } \right\}_{k = 0}^m ,m \in \mathbb{Z}_ + $ , are measures supported on [?1,1] which satisfy Szegö's condition.     

Comparative asymptotics for perturbed orthogonal polynomials

Let and be such systems of orthonormal polynomials on the unit circle that the recurrence coefficients of the perturbed polynomials behave asymptotically like those of . We give, under weak assumptions on the system and the perturbations, comparative asymptotics as for etc., , on the open unit disk and on the circumference mainly off the support of the measure with respect to which the 's are orthonormal. In particular these results apply if the comparative system has a support which consists of several arcs of the unit circumference, as in the case when the recurrence coefficients are (asymptotically) periodic.

     

A shortcut to asymptotics for orthogonal polynomials

We consider the asymptotic behavior of the ratios q_n+1(z)/q_n(z) of polynomials orthonormal with respect to some positive measure μ. Let the recurrence coefficients _n and β_n converge to 0 and , respectively. Then, q_n+1(z)/q_n(z) Φ(z),for n→∞ locally uniformly for , where Φ maps conformally onto the exterior of the unit disc (Nevai (1979)). We provide a new and direct proof for this and some related results due to Nevai, and apply it to convergence acceleration of diagonal Padé approximants.     

On ratio asymptotics for general polynomials

In this paper some characterizations of the ratio asymptotics for general polynomials are given. These results are extensions and improvements of the ratio asymptotics for orthogonal polynomials and are applicable to the ratio asymptotics for polynomials with disturbed nodes.     

Norm asymptotics of orthogonal polynomials for general measures

Letμ be a fixed positive unit Borel measure with infinite support in the unit disk. Acarrier of μ is any Borel subsetB of the support for whichμ(B)=1, and another such measurev iscarrier-related to μ when it has the same carriers asμ. Letp _n (z, v) be the monic orthogonal polynomial of degreen forv. We describe the possible asymptotics for the sequences {(∫|p _n (z,v)|² dv)1/2n} _n≥1 which are associated to the set of measures carrier-related to μ.     

Strong asymptotics for the Pollaczek multiple orthogonal polynomials

The asymptotic properties of multiple orthogonal polynomials with respect to two Pollaczek weights with different parameters are considered. This set of weights is a Nikishin system generated by two measures with unbounded supports; moreover, the second measure is discrete. During the last years, multiple orthogonal polynomials with respect to Nikishin systems of this type have found wide applications in the theory of random matrices. Strong asymptotic formulas for the polynomials under consideration are obtained by means of the matrix Riemann–Hilbert method.     

Relative asymptotics for orthogonal matrix polynomials with respect to a perturbed matrix measure on the unit circle

Given a positive definite matrix measure Ω supported on the unit circle T, then main purpose of this paper is to study the asymptotic behavior of Ln(Ω)Ln(Ω)-1 and Φn(z;Ω)Φn(z;Ω)-1 where Ω(z) = Ω(z) + zδ(z - w); |w| ＞ 1,M is a positive definite matrix and δ is the Dirac matrix measure. Here, Ln ( @ ) means the leading coefficient of the orthonormal matrix polynomials Φn(z; @ ).Finally, we deduce the asymptotic behavior of Φn (w;Ω)Φn (w;Ω) * in the case when M=I.     

Some asymptotics for Sobolev orthogonal polynomials involving Gegenbauer weights

We consider the Sobolev inner product

     

Zeros of polynomials embedded in an orthogonal sequence

Let $\{x_{k,n}\}_{k=1}^n$ and $\{x_{k,n+1}\}_{k=1}^{n+1}$ , n?????, be two given sets of real distinct points with x _1,n?+?1?<?x _1,n ?<?x _2,n?+?1?<?...?<?x _n,n ?<?x _n?+?1,n?+?1. Wendroff (cf. Proc Am Math Soc 12:554?C555, 1961) proved that if $p_n(x)=\displaystyle{\prod\limits_{k=1}^n(x-x_{k,n})}$ and $p_{n+1}(x)=\displaystyle \prod\limits_{k=1}^{n+1}(x-x_{k,n+1})$ then p _n and p _n?+?1 can be embedded in a non-unique infinite monic orthogonal sequence $\{p_n\}_{n=0}^{\infty}$ . We investigate the connection between the zeros of p _n?+?2 and the two coefficients b _n?+?1????? and ?? _n?+?1?>?0, which are chosen arbitrarily, that define p _n?+?2 via the three term recurrence relation $$ p_{n+2}(x)=(x-b_{n+1})p_{n+1}(x)-\lambda_{n+1}p_n(x). $$      

Zeros of orthogonal polynomials on the real line

Let p_n(x) be the orthonormal polynomials associated to a measure dμ of compact support in . If Esupp(dμ), we show there is a δ>0 so that for all n, either p_n or p_n+1 has no zeros in (E−δ,E+δ). If E is an isolated point of supp(μ), we show there is a δ so that for all n, either p_n or p_n+1 has at most one zero in (E−δ,E+δ). We provide an example where the zeros of p_n are dense in a gap of supp(dμ).     

Discrete orthogonal matrix polynomials

At the present time, the theory of orthogonal matrix polynomials is an active area of mathematics and exhibits a promising future. However, the discrete case has been completely forgotten. In this note we introduce the notion of discrete orthogonal matrix polynomials, and show some algebraic properties. In particular, we study a matrix version of the usual Meixner polynomials.     

Strong asymptotics for Sobolev orthogonal polynomials in the complex plane

We obtain the strong asymptotics for the sequence of monic polynomials minimizing the norm

     

Weak convergence for orthogonal matrix polynomials

We study the weak convergence of orthogonal matrix polynomials under some conditions on the asymptotic behaviour of the coefficients in the three-term recurrence relation.     

Zeros of Sobolev orthogonal polynomials following from coherent pairs

Let {S_n^λ} denote the monic orthogonal polynomial sequence with respect to the Sobolev inner product$\text{〈f,g〉}{}_{\mathrm{S}}\text{=}\text{∫}\text{−∞}\text{∞}\text{fg}\text{d}\text{ψ}{}_0\text{+λ}\text{∫}\text{−∞}\text{∞}\text{f′g′}\text{d}\text{ψ}{}_1\text{,}$where {dψ₀,dψ₁} is a so-called coherent pair and λ>0. Then S_n^λ has n different, real zeros. The position of these zeros with respect to the zeros of other orthogonal polynomials (in particular Laguerre and Jacobi polynomials) is investigated. Coherent pairs are found where the zeros of S_n−1^λ separate the zeros of S_n^λ.