DOI: http://dx.doi.org/10.18273/revint.v34n1-2016003

On a finite moment perturbation of linear
functionals and the inverse SzegŐ
transformation

EDINSON FUENTES^a*, LUIS E. GARZA^b

^a Universidad Pedagógica y Tecnológica de Colombia, Escuela de Matemáticas y
Estadística, Tunja, Colombia.
^b Universidad de Colima, Facultad de Ciencias, Colima, México.

Abstract. Given a sequence of moments {c_n}n∈ℤ associated with an Hermitian linear functional L defined in the space of Laurent polynomials, we study a new functional L_Ω which is a perturbationof L in such a way that a finite number of moments are perturbed. Necessary and sufficient conditions are given for the regularity of L_Ω, and a connection formula between the corresponding families of orthogonal polynomials is obtained. On the other hand, assuming L_Ω is positive definite, the perturbation is analyzed through the inverse SzegŐ transformation.

Keywords: Orthogonal polynomials on the unit circle, perturbation of moments, inverse SzegŐ transformation.
MSC2010: 42C05, 33C45, 33D45, 33C47.

Sobre una perturbación finita de momentos de un
funcional lineal y la transformación inversa de SzegŐ

Resumen. Dada una sucesión de momentos {c_n}n∈ℤ asociada a un funcional lineal hermitiano L definido en el espacio de los polinomios de Laurent, estudiamos un nuevo funcional L_Ω que consiste en una perturbación de L de tal forma que se perturba un número finito de momentos de la sucesión. Se encuentran condiciones necesarias y suficientes para la regularidad de L_Ω, y se obtiene una fórmula de conexión que relaciona las familias de polinomios ortogonales correspondientes. Por otro lado, suponiendo que L_Ω es definido positivo, se analiza la perturbación mediante de la transformación inversa de SzegŐ.

Palabras clave: Polinomios ortogonales en la circunferencia unidad, perturbación de momentos, transformación de SzegŐ inversa.

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^*Email: edinson.fuentes@uptc.edu.co
Received: 13 October 2015, Accepted: 03 February 2016.
To cite this article: E. Fuentes, L.E. Garza, On a finite moment perturbation of linear functionals and the inverse SzegŐ transformation, Rev. Integr. Temas Mat. 34 (2016), No. 1, 39-58.