ON A UVAROV TYPE MODIFICATION OF ORTHOGONAL POLYNOMIALS ON THE UNIT BALL
MIGUEL A. PI ˜NAR
Abstract
In the present work, we study orthogonal polynomials with respect to a Uvarov type modification of the classical inner product on the unit ball. Namely, we con- sider the inner product
hf, giλµ= 1 ωµ
Z
Bd
f (x)g(x)Wµ(x)dx + λ σd
Z
Sd−1
f (ξ)g(ξ)dσ(ξ),
where ωµ is a normalizing constant, dσ denotes the surface measure and σd−1 denotes the surface area. Here λ > 0 represents a mass uniformly distributed over the sphere.
Using spherical polar coordinates, we shall construct a sequence of mutually orthogonal polynomials with respect to h·, ·iλµ, which depends on a family orthog- onal polynomials in one variable. These polynomials are orthogonal with respect to a Uvarov modification of a varying Jacobi measure and therefore they can be expressed in terms of Jacobi polynomials. Explicit representations tor the polyno- mials, the norms and the kernels will be obtained.
A very interesting problem in the theory of multivariate orthogonal polynomials is that of finding asymptotic estimates for the Christoffel functions. These estimates are useful in the study of the convergence of the Fourier series. Asymptotics for Christoffel functions associated to the classical orthogonal polynomials on the ball were obtained by Y. Xu in 1996 (see [2]). Recently, more general results on the asymptotic behaviour of the Christoffel functions for some kind of regular measures were established by Kro´o and Lubinsky [1].
Since our system of orthogonal polynomials does not fit into the above mentioned context, the asymptotic of the Christoffel functions deserves special attention. Not surprisingly, our results show that in any compact subset of the interior of the unit ball Christoffel functions for the Uvarov type modification behave exactly as in the classical case. On the sphere the situation is quite different and we can perceive the influence of the mass λ.
References
[1] A. Kro´o and D. S. Lubinsky. Christoffel functions and universality in the bulk for multivariate orthogonal polynomials. Canad. J. Math., 65(3):600–620, 2013.
[2] Y. Xu. Asymptotics for orthogonal polynomials and Christoffel functions on a ball. Methods Appl. Anal., 3(2):257–272, 1996.
(M. A. Pi˜nar) Departamento de Matem´atica Aplicada, Universidad de Granada, 18071 Granada, Spain
E-mail address: [email protected]
