Tight frames of orthogonal polynomials on the simplex

Abstract

The orthogonal polynomials of degree k on the triangle are a finite dimensional inner product space which is invariant under the unitary action of the symmetry group G of the triangle (the dihedral group of order 6) given by

$$g\cdot f= f\circ g^{-1}, $$

i.e., a G-invariant space (see §10.10).

Appendices

Notes

The idea of using finite tight frame expansions for spaces of multivariate orthogonal polynomials (not to be confused with multiple orthogonal polynomials [MFVA16]) appeared independently in [Ros99] and [XW01],  [PW02]. A detailed account of the multivariate orthogonal polynomials, which includes the systems of Appell and Prorial, is given in [DX01]. The presentation in terms of the Bernstein–Durrmeyer operator $M_n^\nu$ that is given here is adapted from [RW04],  [Wal06]. There are similar expansions for the multivariate Hahn and continuous Hahn polynomials [RW04], and for the multivariate orthogonal polynomials for a radially symmetric weight (see Chapter 16).

Tight frames allow for optimal expansions for spaces of multivariate orthogonal polynomials for specific weights, e.g., see [Dun87] and §10.10, §10.14.

Exercises

15.1.

Let R be the degree raising operator given by (15.4), and $R_\nu ^*$ be its adjoint as given by (15.6). Show that the j-th power of $R_\nu ^*$ is given by (15.7), i.e.,

$$((R_\nu ^*)^j b)_\beta = \sum _{|\gamma |=j}{(\beta +\nu )_{\gamma }\over (|\beta |+1)_j} {j\atopwithdelims ()\gamma } b_{\beta +\gamma } , \qquad b:\varDelta _n\rightarrow \mathbb {R},\quad 0\le j\le n. $$