Asymptotic Behaviour of the Christoffel Functions on the Unit Ball in the Presence of a Mass on the Sphere
- 64 Downloads
We present a family of mutually orthogonal polynomials on the unit ball with respect to an inner product which includes a mass uniformly distributed on the sphere. First, connection formulas relating these multivariate orthogonal polynomials and the classical ball polynomials are obtained. Then, using the representation formula for these polynomials in terms of spherical harmonics analytic properties will be deduced. Finally, we analyse the asymptotic behaviour of the Christoffel functions.
KeywordsMultivariate orthogonal polynomials unit ball Uvarov modification
Mathematics Subject Classification33C50 42C10
The authors thank MINECO of Spain and the European Regional Development Fund (ERDF) through Grant MTM2014-53171-P, and Junta de Andalucía Grant P11-FQM-7276 and research group FQM–384.
- 1.Dai, F., Xu, Y.: Approximation Theory and Harmonic Analysis on Spheres and Balls. Springer Monographs in Mathematics. Springer, New York (2013)Google Scholar
- 9.Martínez, C., Piñar, M.A.: Orthogonal polynomials on the unit ball and fourth-order partial differential equations. SIGMA Symmetry Integrability Geom. Methods Appl., 12:Paper No. 20, 11 (2016)Google Scholar
- 10.Nevai, P.: Orthogonal polynomials. Mem. Am. Math. Soc., 18(213) (1979)Google Scholar
- 11.Szegő, G.: Orthogonal polynomials. American Mathematical Society, Providence, R.I., fourth edition, American Mathematical Society, Colloquium Publications, Vol. XXIII (1975)Google Scholar
- 13.Wimp, J.: Sequence Transformations and Their Applications, Volume 154 of Mathematics in Science and Engineering. Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London (1981)Google Scholar