期刊
CONSTRUCTIVE APPROXIMATION
卷 37, 期 3, 页码 383-453出版社
SPRINGER
DOI: 10.1007/s00365-013-9184-4
关键词
Multiple orthogonal polynomial; Nikishin system; Banded Hessenberg matrix; Block Toeplitz matrix; Riemann-Hilbert matrix; Generalized Poincare theorem; Ratio asymptotics; Vector equilibrium problem; Interlacing; Totally positive matrix
类别
资金
- Postdoctoral Fellows of the Fund for Scientific Research-Flanders (FWO), Belgium
We study monic polynomials Q (n) (x) generated by a high-order three-term recursion xQ (n) (x)=Q (n+1)(x)+a (n-p) Q (n-p) (x) with arbitrary pa parts per thousand yen1 and a (n) > 0 for all n. The recursion is encoded by a two-diagonal Hessenberg operator H. One of our main results is that, for periodic coefficients a (n) and under certain conditions, the Q (n) are multiple orthogonal polynomials with respect to a Nikishin system of orthogonality measures supported on star-like sets in the complex plane. This improves a recent result of Aptekarev-Kalyagin-Saff, where a formal connection with Nikishin systems was obtained in the case when for some a > 0. An important tool in this paper is the study of 'Riemann-Hilbert minors', or equivalently, the 'generalized eigenvalues' of the Hessenberg matrix H. We prove interlacing relations for the generalized eigenvalues by using totally positive matrices. In the case of asymptotically periodic coefficients a (n) , we find weak and ratio asymptotics for the Riemann-Hilbert minors and we obtain a connection with a vector equilibrium problem. We anticipate that in the future, the study of Riemann-Hilbert minors may prove useful for more general classes of multiple orthogonal polynomials.
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