Ratio asymptotics and weak asymptotic measures for orthogonal polynomials on the real line
نویسنده
چکیده
We study ratio asymptotics, that is, existence of the limit of Pnþ1ðzÞ=PnðzÞ (Pn 1⁄4 monic orthogonal polynomial) and the existence of weak limits of pn dm ðpn 1⁄4 Pn=jjPnjjÞ as n-N for orthogonal polynomials on the real line. We show existence of ratio asymptotics at a single z0 with Imðz0Þa0 implies dm is in a Nevai class (i.e., an-a and bn-b where an; bn are the offdiagonal and diagonal Jacobi parameters). For m’s with bounded support, we prove pn dm has a weak limit if and only if lim bn; lim a2n; and lim a2nþ1 all exist. In both cases, we write down the limits explicitly. r 2003 Elsevier Inc. All rights reserved. MSC: 42C05; 47B36; 81Q10
منابع مشابه
Strong asymptotics of orthogonal polynomials with varying measures and Hermite–Pad e approximants
The strong asymptotic behaviour of orthogonal polynomials with respect to a general class of varying measures is given for the case of the unit circle and the real line. These results are used to obtain certain asymptotic relations for the polynomials involved in the construction of Hermite–Pad e approximants of a Nikishin system of functions. c © 1998 Elsevier Science B.V. All rights reserved....
متن کاملLogarithmic asymptotics of contracted Sobolev extremal polynomials on the real line
For a wide class of Sobolev type norms with respect to measures with unbounded support on the real line, the contracted zero distribution and the logarithmic asymptotic of the corresponding re-scaled Sobolev orthogonal polynomials is given.
متن کاملRelative Asymptotics for Polynomials Orthogonal with Respect to a Discrete Sobolev Inner Product
We investigate the asymptotic properties of orthogonal polynomials for a class of inner products including the discrete Sobolev inner products 〈h, g〉 = ∫ hg dμ+ ∑m j=1 Nj i=0 Mj,ih (cj)g (cj), where μ is a certain type of complex measure on the real line, and cj are complex numbers in the complement of supp(μ). The Sobolev orthogonal polynomials are compared with the orthogonal polynomials corr...
متن کاملOrthogonal polynomials for a class of measures with discrete rotational symmetries in the complex plane
We obtain the strong asymptotics of polynomials pn(λ), λ ∈ C, orthogonal with respect to measures in the complex plane of the form e 2s−tλs−tλs)dA(λ), where s is a positive integer, t is a complex parameter and dA stands for the area measure in the plane. Such problem has its origin from normal matrix models. We study the asymptotic behaviour of pn(λ) in the limit n,N → ∞ in such a way that n/N...
متن کاملStrong asymptotics for Gegenbauer-Sobolev orthogonal polynomials
We study the asymptotic behaviour of the monic orthogonal polynomials with respect to the Gegenbauer-Sobolev inner product (f, g)S = 〈f, g〉 + λ〈f ′, g′〉 where 〈f, g〉 = ∫ 1 −1 f(x)g(x)(1 − x 2)α−1/2dx with α > −1/2 and λ > 0. The asymptotics of the zeros and norms of these polynomials is also established. The study of the orthogonal polynomials with respect to the inner products that involve der...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
- Journal of Approximation Theory
دوره 126 شماره
صفحات -
تاریخ انتشار 2004