Favard, Baxter, Geronimus, Rakhmanov, Szegö and the strong Szegö theorems for orthogonal trigonometric polynomials

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.

单位圆上Laurent多重正交多项式的零点

Zeros of Laurent multiple orthogonal polynomials on the unit circle

研究单位圆上Laurent多重正交多项式的两种不同定义，证明第一种定义下所有零点严格位于复单位盘内，第二种定义下所有索引形式的正常性，从而可应用Szegő映射和Geronimus关系。