an:06004440
Zbl 1247.33023
Waldron, Shayne
Recursive three-term recurrence relations for the Jacobi polynomials on a triangle
EN
Constr. Approx. 33, No. 3, 405-424 (2011).
0176-4276 1432-0940
2011
j
33C50 42C05 33C65 33C80
(recursive) three-term recurrence relations; multivariate Jacobi polynomials; Legendre polynomials on a triangle; barycentric coordinates; symmetry group
Given a suitable weight on \(\mathbb{R}^d\), there exist many (recursive) three-term recurrence relations for the corresponding multivariate orthogonal polynomials. These can be obtained by calculating pseudoinverses of a sequence of matrices. The author gives an explicit recursive three-term recurrence relation for the multivariate Jacobi polynomials on a simplex. This formula is obtained by seeking the best possible three-term recurrence relation. This defines corresponding linear maps, which have the same symmetries as the spaces of Jacobi polynomials on which they are defined. The key idea behind this formula is that some Jacobi polynomials on a simplex can be viewed as univariate Jacobi polynomials, and for these the recurrence relation reduces to the univariate three-term recurrence relation.
Roelof Koekoek (Delft)