Orthogonal polynomials and measures on the unit circle. The Geronimus transformations

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In this paper we analyze a perturbation of a nontrivial positive measure supported on the unit circle. This perturbation is the inverse of the Christoffel transformation and is called the Geronimus transformation. We study the corresponding sequences of monic orthogonal polynomials as well as the connection between the associated Hessenberg matrices. Finally, we show an example of this kind of transformation.
12 pages, no figures.-- MSC2000 codes: 42C05, 15A23.-- Published in Special Issue dedicated to William B. Gragg on the occasion of his 70th birthday.
Elsevier
Journal of Computational and Applied Mathematics, 2010, vol. 233, n. 5, p. 1220-1231
The work of the first author (L. Garza) has been supported by a grant from the Universidad Autónoma de Tamaulipas. The work of the second author (J. Hernández) has been supported by a grant from the Fundación Universidad Carlos III de Madrid. The research of the third author (F. Marcellán) has been supported by Dirección General de Investigación, Ministerio de Educación y Ciencia of Spain, grant MTM 2006-13000-C03-02.

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Orthogonal polynomials and measures on the unit circle. The Geronimus transformations
L. Garza, J. Hernández, F. Marcellán
Departamento de Matemáticas, Universidad Carlos III de Madrid, Avenida de la Universidad 30, 28911, Leganés, Spain. To professor William B. Gragg with occasion of his70thbirthday
Abstract
In this paper we analyze a perturbation of a nontrivial positive measure supported on the unit circle. This perturbation is the inverse of the Christoel transformation and is called the Geronimus transformation. We study the corresponding sequences of monic orthogonal polynomials as well as the connection between the associated Hessenberg matrices. Finally, we show an example of this kind of transformation.
Key words:Measures on the unit circle, orthogonal polynomials, spectral transformations, Hessenberg matrices.
2000AMS classification:42C05, 15A23.
1 Introduction
The study of orthogonal polynomials with respect to a nontrivial positive Borel measure supported on the unit circleT={zC:|z|=1}was started by G. Szego˝ in several papers published from 1915 to 1925 (see [19]). Later on Y. Geronimus [4] extended this theory to a more general situation.
Ifυis a linear functional in the linear spaceΛof the Laurent polynomials (Λ =span{zn}nZ) such thatυis Hermitian, i. e.cn=hυ,zni=hυ,zni=c¯n, nZ, then a bilinear functional associated withυcan be introduced in the linear Email addresses:@aauagno.uxm.tdel(L. Garza),.htam@etse.m3cubenijh(J. Hernández),sucg..e3maacropminc@(F. Marcellán).
PreprintsubmittedtoElsevier
13November2007
spacePof polynomials with complex coecient as follows (p(z),q(z))υ=Dυ,p(z) ¯q(z1)E
wherep,qP.
(1)
The Gram matrix associated with this bilinear functional in terms of the canonical basis{zn}n>0ofPis c0c1∙ ∙ ∙cn∙ ∙ ∙c1c0∙ ∙ ∙cn1∙ ∙ ∙ T=. .....,(2) cncn+1∙ ∙ ∙c0∙ ∙ ∙ . . ....a Toeplitz matrix [8].
The linear functional is said to be quasi-definite if the principal leading submatrices ofTare non-singular. If such matrices have positive determinant, then the linear functional is said to be positive definite. Every positive definite linear functional has an integral representation Z
hυ,p(z)i=p(z)dµ(z), T
(3)
whereµsupported on the unit circle (see [4],is a nontrivial positive Borel measure [8], [11], [17]).
Ifυis a quasi-definite linear functional then a unique sequence of monic polyno-mials{Pn}n>0such that = (Pn,Pm)υknδn,m,(4) can be introduced, wherekn,0 for everyn>0. It is said to be the monic orthog-onal polynomial sequence associated withυ.
This polynomial sequence satisfies two equivalent recurrence relations due to G. Szeg˝o(see[4],[8],[17],[19])
Pn+1(z)=zPn(z)+Pn+1(0)Pn(z),(5) Pn+1(z)=1− |Pn+1(0)|2zPn(z)+Pn+1(0)Pn+1(z),(6) P¯n(z1s the the forward and backward recurrences, respectively, wherePn(z)=zn) i so-called reversed polynomial. On the other hand, from (5) and (6) we deduce
n+1 zPn(z)=Xλn,jPj(z), j=0
2
(7)