Orthogonal polynomials on the sphere with octahedral symmetry

Abstract

Abstract.For any finite reflection group G acting on RN there is a family of G-invariant measures (h2 du, where A is a certain product of linear functions whose zero-sets are the reflecting hyperplanes for G) on the unit sphere and an associated partial differential operator ( Lhf ■ = A( fh) -/Aft; A is the Laplacian). Analogously to spherical harmonics, there is an orthogonal (with respect to h2 du) decomposition of homogeneous polynomials, that is, if p is of degree n then l«/2]

Reflection groups in analysis and applications

“…Using Green's theorem it is straightforward to show that S N−1 f gh 2 κ dm = 0 whenever f , g are homogeneous polynomials of different degrees and Δ κ f = 0 = Δ κ g. However if κ(v) = 0 for all v ∈ R then the hypotheses on f and g imply that they are W -invariant (see [9,Th. 2.4]).…”

Nonsymmetric Jack polynomials and integral kernels

“…In ref. [9] Dunkl has extended the theory of [8] to the non-symmetric case, and this allows the considerations of [5] to be similarly extended.…”

Orthogonal Polynomials and Cubature Formulae on Spheres and on Balls

“…They are orthogonal polynomials with respect to the surface measure dω on S d . The theory of h-harmonics has been established recently by Dunkl (see [3][4][5]). For a nonzero vector v ∈ R d+1 define the reflection σ v by…”

“…The main results will be established for a large class of measures. Orthogonal polynomials on the sphere with respect to a measure other than the surface measure have been studied only recently (see [3,4,5,8,25] and the references there). The most important development has been a theory developed by Dunkl for measures invariant under a finite reflection group, in which the role of Laplacian operator is replaced by a differential-difference operator in the commutative algebra generated by a family of commuting first order differential-difference operators (Dunkl's operators).…”