The asymptotics of the kernel functions associated to orthogonal polynomials in several variables on the unit ball

We consider a family of orthogonal polynomials in several variables with respect to a Sobolev-type inner product, obtained from adding a gradient operator of order j, evaluated in a fixed point to a standard inner product. We study explicit relations between the Sobolev-type polynomials and the stand...

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Bibliographic Details
Published in:Boletín de matemáticas Vol. 22; no. 1; pp. 39 - 53
Main Authors: Dueñas, Herbert, Gómez Blanco, Wilmer Merardo
Format: Journal Article
Language:English
Published: 2015
Subjects:
Asymptotics behavior
comportamiento asintótico
Orthogonal polynomials in several variables
Polinomios ortogonales en varias variables
producto interno de tipo Sobolev
Sobolev inner products
ISSN:0120-0380
Online Access:Get full text
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Summary:We consider a family of orthogonal polynomials in several variables with respect to a Sobolev-type inner product, obtained from adding a gradient operator of order j, evaluated in a fixed point to a standard inner product. We study explicit relations between the Sobolev-type polynomials and the standard polynomials, among the kernel functions associated to the Sobolev-type polynomials and the kernel functions associated to the standard polynomials. In addition, an example for a particular choice of a classical measure σ ∈ Rd is analyzed. Finally, we obtain the asymptotics of the some derivatives of the kernel functions evaluated in some points of the unit ball in d variables. Consideramos una familia de polinomios ortogonales en varias variables con respecto a un producto interno de tipo Sobolev, el cual se obtiene al adicionar a un producto interno estándar un operador gradiente de orden j, evaluado en un punto fijo. Estudiamos relaciones entre los polinomios de tipo Sobolev y los polinomios estándar, como relaciones entre el núcleo asociado a los polinomios de tipo Sobolev y el núcleo de los polinomios estándar. Adicionalmente, estudiamos un caso particular de una medida σ ∈ Rd Finalmente, se obtienen los comportamientos asintóticos de las derivadas del núcleo evaluadas en puntos de la bola unidad en d variables.
ISSN:0120-0380