Szego polynomials: a view from the Riemann-Hilbert window

Orthogonal trigonometric polynomials from Riemann-Hilbert view

In this work, some theorems are established for orthogonal trigonometric polynomials (OTP) including Favard, Baxter, Geronimus, Rakhmanov, Szegö and the strong Szegö theorems which are important in the theory of orthogonal polynomials on the unit circle (OPUC). All these results are based on the mutual representation theorem of OPUC and OTP which deduced by a Riemann-Hilbert problem simultaneously characterizing them. In addition, Szegö recursions, four-term recurrences and some new identities for OPUC and OTP are also obtained by using their Riemann-Hilbert characterizations respectively.

Szegő orthogonal polynomials with respect to an analytic weight: canonical representation and strong asymptotics

We provide a representation in terms of certain canonical functions for a sequence of polynomials orthogonal with respect to a weight that is strictly positive and analytic on the unit circle. These formulas yield a complete asymptotic expansion for these polynomials, valid uniformly in the whole complex plane. As a consequence, we obtain some results about the distribution of zeros of these polynomials. The main technique is the steepest descent analysis of Deift and Zhou, based on the matrix Riemann-Hilbert characterization proposed by Fokas, Its and Kitaev.

Matrix biorthogonal polynomials in the unit circle: Riemann-Hilbert problem and matrix discrete Painleve II system

Matrix Szego biorthogonal polynomials for quasi-definite matrices of measures are studied. For matrices of Holder weights a Riemann-Hilbert problem is uniquely solved in terms of the matrix Szego polynomials and its Cauchy transforms. The Riemann-Hilbert problem is given as an appropriate framework for the discussion of the Szego matrix and the associated Szego recursion relations for the matrix orthogonal polynomials and its Cauchy transforms. Pearson type differential systems characterizing the matrix of weights are studied. These are linear systems of ordinary differential equations which a…

多割下的矩阵Szegő函数与矩阵正交多项式

Matrix Szegő Function and Matrix Orthogonal Polynomials for Multiple Cuts

本文针对多割下的矩阵权函数，构造其矩阵Szegő分解，通过Deift-Zhou最速下降分析得到矩阵值正交多项式的强渐近式，扩展了相关学者的已有研究。

The Riemann-Hilbert approach to strong asymptotics for orthogonal polynomials on [-1,1]

We consider polynomials that are orthogonal on $[-1,1]$ with respect to a modified Jacobi weight $(1-x)^α(1+x)^βh(x)$, with $α,β>-1$ and $h$ real analytic and stricly positive on $[-1,1]$. We obtain full asymptotic expansions for the monic and orthonormal polynomials outside the interval $[-1,1]$, for the recurrence coefficients and for the leading coefficients of the orthonormal polynomials. We also deduce asymptotic behavior for the Hankel determinants. For the asymptotic analysis we use the steepest descent technique for Riemann--Hilbert problems developed by Deift and Zhou, and applied to …

紧致黎曼曲面上的极值函数与维德姆因子

Extremal Functions and Widom Factors on Compact Riemann Surfaces

研究亏格\(g>0\)紧致黎曼曲面上切比雪夫极值问题，考虑特定容许亚纯函数，证明相关常数收敛及不等式。利用柯西核研究维德姆因子渐近性，揭示无类似法伯型渐近性，最后在亏格\(1\)时构造了明确例子。