Analysis of perturbations of moments
associated with orthogonality linear
functionals through the Szegő transformation
EDINSON FUENTESa*, LUIS E. GARZAb,c
a Universidad Pedagígica y Tecnolígica de Colombia, Escuela de Matemáticas y Estadística, Tunja, Colombia.
b Universidad Nacional de Colombia, Departamento de Matemáticas, Bogotá, Colombia.
c Universidad de Colima, Facultad de Ciencias, Colima, México.
Abstract. In this paper, we consider perturbations to a sequence of moments associated with an orthogonality linear functional that is represented by a positive measure supported in [-1, 1]. In particular, given a perturbation to such a measure on the real line, we analyze the perturbation obtained on the corresponding measure on the unit circle, when both measures are related through the Szegő transformation. A similar perturbation is analyzed through the inverse Szegő transformation. In both cases, we show that the applied perturbation can be expressed in terms of the singular part of the measures, and also in terms of the corresponding sequences of moments.
Keywords: Orthogonal polynomials, Stieltjes and Carathéodory functions, Hankel and Toeplitz matrices, Szegő transformation. ]]>
MSC2010: 42C05, 33C45, 33D45, 33C47.Análisis de perturbaciones de momentos asociados a
funcionales de ortogonalidad a través de la
transformaciín de Szegő
Resumen. En el presente trabajo, analizamos las perturbaciones a una sucesiín de momentos asociada a un funcional lineal de ortogonalidad que se representa por una medida positiva con soporte en [-1, 1]. En particular, dada una cierta perturbaciín a dicha medida en la recta real, analizamos la perturbaciín obtenida en la correspondiente medida en la circunferencia unidad, cuando dichas medidas están relacionadas por la transformaciín de Szegő. También se analiza una perturbaciín similar a través de la transformaciín inversa de Szegő. En ambos casos, se muestra que la perturbaciín aplicada puede ser expresada en términos de la parte singular de las medidas, y también a través de las correspondientes sucesiones de momentos.
Palabras clave: Polinomios ortogonales, funciones de Stieltjes y Carathéodory, matrices de Hankel y Toeplitz, transformaciín de Szegő.
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Referencias
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*E-mail: edinson.fuentes@uptc.edu.co.
Received: 3 February 2015, Accepted: 15 April 2015.
To cite this article: E. Fuentes, L.E. Garza, Analysis of perturbations of moments associated with orthogonality linear functionals through the Szegő transformation, Rev. Integr. Temas Mat. 33 (2015), no. 1, 61-82.