Orthogonal polynomials on the unit circle via a polynomial mapping on the real line
- 作者：J. Petronilho
- 关键词：Orthogonal polynomials ; Unit circle ; Polynomial mappings ; Verblunsky coefficients ; Recurrence relations ; Stieltjes transforms ; Carathé ; odory functions ; Borel measures
- 刊名：Journal of Computational and Applied Mathematics
- 出版年：2008
- 期刊代码：72_03770427
- 类别：cp
- 出版时间：15 June 2008
- 卷：216
- 期：1
- 页码：98-127
- 文件大小：381 K
摘要
Let {Φn}n
0 be a sequence of monic orthogonal polynomials on the unit circle (OPUC) with respect to a symmetric and finite positive Borel measure dμ on [0,2π] and let -1,
0,1,2,… be the associated sequence of Verblunsky coefficients. In this paper we study the sequence 
of monic OPUC whose sequence of Verblunsky coefficients is
where b1,b2,…,bN-1 are N-1 fixed real numbers such that bj
(-1,1) for all j=1,2,…,N-1, so that 
is also orthogonal with respect to a symmetric and finite positive Borel measure 
on the unit circle. We show that the sequences of monic orthogonal polynomials on the real line (OPRL) corresponding to {Φn}n0 and 
(by Szegö's transformation) are related by some polynomial mapping, giving rise to a one-to-one correspondence between the monic OPUC 
on the unit circle and a pair of monic OPRL on (a subset of) the interval [-1,1]. In particular we prove that
supported on (a subset of) the union of 2N intervals contained in [0,2π] such that any two of these intervals have at most one common point, and where, up to an affine change in the variable, ζN-1 and cos
N are algebraic polynomials in cosθ of degrees N-1 and N (respectively) defined only in terms of 0,b1,…,bN-1. This measure induces a measure on the unit circle supported on the union of 2N arcs, pairwise symmetric with respect to the real axis. The restriction to symmetric measures (or real Verblunsky coefficients) is needed in order that Szegö's transformation may be applicable.
0 be a sequence of monic orthogonal polynomials on the unit circle (OPUC) with respect to a symmetric and finite positive Borel measure dμ on [0,2π] and let -1,0,1,2,… be the associated sequence of Verblunsky coefficients. In this paper we study the sequence where b1,b2,…,bN-1 are N-1 fixed real numbers such that bj
(-1,1) for all j=1,2,…,N-1, so that 0 and supported on (a subset of) the union of 2N intervals contained in [0,2π] such that any two of these intervals have at most one common point, and where, up to an affine change in the variable, ζN-1 and cos
N are algebraic polynomials in cosθ of degrees N-1 and N (respectively) defined only in terms of 0,b1,…,bN-1. This measure induces a measure on the unit circle supported on the union of 2N arcs, pairwise symmetric with respect to the real axis. The restriction to symmetric measures (or real Verblunsky coefficients) is needed in order that Szegö's transformation may be applicable.