Laurent 多项式及其零点

本文给出了测度dψ为强分布的一个必要条件,并得到了dψ为强分布时的Laurent多项式最大零点的一个表示。     

Blumenthal''s Theorem for Laurent Orthogonal Polynomials

We investigate polynomials satisfying a three-term recurrence relation of the form B_n(x)=(x−β_n)B_n−1(x)−α_nxB_n−2(x), with positive recurrence coefficients α_n+1,β_n (n=1,2,…). We show that the zeros are eigenvalues of a structured Hessenberg matrix and give the left and right eigenvectors of this matrix, from which we deduce Laurent orthogonality and the Gaussian quadrature formula. We analyse in more detail the case where α_n→α and β_n→β and show that the zeros of B_n are dense on an interval and that the support of the Laurent orthogonality measure is equal to this interval and a set which is at most denumerable with accumulation points (if any) at the endpoints of the interval. This result is the Laurent version of Blumenthal's theorem for orthogonal polynomials.     

Mixed Series of Chebyshev Polynomials Orthogonal on a Uniform Grid

We construct an expansion of a discrete function in the form of a mixed series of Chebyshev polynomials. We obtain estimates of the approximation error of the function and its derivatives.     

Orthogonal Polynomials for Modified Chebyshev Measure of the First Kind

Given numbers \({n,s \in \mathbb{N}}\), \({n \geq 2}\), and the \({n}\)th-degree monic Chebyshev polynomial of the first kind \({\widehat T_n(x)}\), the polynomial system “induced” by \({\widehat T_n(x)}\) is the system of orthogonal polynomials \({\{p_{k}^{n,s} \}}\) corresponding to the modified measure \({d \sigma^{n,s}(x)=\widehat T^{2s}_n(x) d\sigma(x)}\), where \({d\sigma(x)=1/\sqrt{1-x^{2}}dx}\) is the Chebyshev measure of the first kind. Here we are concerned with the problem of determining the coefficients in the three-term recurrence relation for the polynomials \({p^{n,s}_{k}}\). The desired coefficients are obtained analytically in a closed form.     

Asymptotics of Laurent Polynomials of Odd Degree Orthogonal with Respect to Varying Exponential Weights

Let denote the linear space over spanned by . Define the (real) inner product , where V satisfies: (i) V is real analytic on ; (ii) ; and (iii) . Orthogonalisation of the (ordered) base with respect to yields the even degree and odd degree orthonormal Laurent polynomials , and . Define the even degree and odd degree monic orthogonal Laurent polynomials: and . Asymptotics in the double-scaling limit such that of (in the entire complex plane), , and (in the entire complex plane) are obtained by formulating the odd degree monic orthogonal Laurent polynomial problem as a matrix Riemann-Hilbert problem on , and then extracting the large-n behaviour by applying the non-linear steepest-descent method introduced in [1] and further developed in [2],[3].     

Chebyshev Polynomials and Integer Coefficients

Mathematical Notes - Generalized Chebyshev polynomials are introduced and studied in this paper. They are applied to obtain a lower bound for the sup-norm on the closed interval for nonzero...     

Sato-Tate Conjectures and Chebyshev Polynomials

Results of Shahidi on the analytic continuation of certain L-functions have been used by Serre to obtain partial information on the distribution of Sato-Tate angles of the Ramanujan function. Employing the Christoffel numbers associated to the orthogonal polynomials used by Serre we derive upper bounds on the lower density of primes with small angles. We also make effective his lower bounds on the upper density of such primes. Assuming that more analytic information is available, we sharpen similar estimates of Ram Murty, and, using the Christoffel numbers attached to the second order Chebyshev polynomials, derive general bounds on the discrepancy between the empirical angles distribution and the Sato-Tate distribution.     

Embedding Distributions and Chebyshev Polynomials

The history of genus distributions began with J. Gross et?al. in 1980s. Since then, a lot of study has given to this parameter, and the explicit formulas are obtained for various kinds of graphs. In this paper, we find a new usage of Chebyshev polynomials in the study of genus distribution, using the overlap matrix, we obtain homogeneous recurrence relation for rank distribution polynomial, which can be solved in terms of Chebyshev polynomials of the second kind. The method here can find explicit formula for embedding distribution of some other graphs. As an application, the well known genus distributions of closed-end ladders and cobblestone paths (Furst et?al. in J Combin Ser B 46:22–36, 1989) are derived. The explicit formula for non-orientable embedding distributions of closed-end ladders and cobblestone paths are also obtained.     

On Integer Chebyshev Polynomials

We are concerned with the problem of minimizing the supremum norm on of a nonzero polynomial of degree at most with integer coefficients. We use the structure of such polynomials to derive an efficient algorithm for computing them. We give a table of these polynomials for degree up to and use a value from this table to answer an open problem due to P. Borwein and T. Erdélyi and improve a lower bound due to Flammang et al.

     

Dedekind和及第一类Chebyshev多项式

本文利用分析方法、Dedekind和及第一类Chebyshev多项式的算术性质,研究了一类关于Dedekind和及第一类Chebyshev多项式混合均值的渐近估计问题,并得到了一个较强的渐近公式.     

Strong Asymptotics of Laguerre-Type Orthogonal Polynomials and Applications in Random Matrix Theory

We consider polynomials orthogonal on [0,∞) with respect to Laguerre-type weights w(x) = x^α e^-Q(x), where α > -1 and where Q denotes a polynomial with positive leading coefficient. The main purpose of this paper is to determine Plancherel-Rotach-type asymptotics in the entire complex plane for the orthonormal polynomials with respect to w, as well as asymptotics of the corresponding recurrence coefficients and of the leading coefficients of the orthonormal polynomials. As an application we will use these asymptotics to prove universality results in random matrix theory. We will prove our results by using the characterization of orthogonal polynomials via a 2 × 2 matrix valued Riemann--Hilbert problem, due to Fokas, Its, and Kitaev, together with an application of the Deift-Zhou steepest descent method to analyze the Riemann-Hilbert problem asymptotically.     

On the Strong Asymptotics for Sobolev Orthogonal Polynomials on the Circle

In the present paper we prove Szegő's asymptotic theorem for the orthogonal polynomials with respect to a Sobolev inner product of the following type:

Orthogonal Polynomials Associated with Related Measures and Sobolev Orthogonal Polynomials

Connection between two sequences of orthogonal polynomials, where the associated measures are related to each other by a first degree polynomial multiplication (or division), are looked at. The results are applied to obtain information regarding Sobolev orthogonal polynomials associated with certain pairs of measures.