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
i.e., a G-invariant space (see §10.10).
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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.,
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Waldron, S.F.D. (2018). Tight frames of orthogonal polynomials on the simplex. In: An Introduction to Finite Tight Frames. Applied and Numerical Harmonic Analysis. Birkhäuser, New York, NY. https://doi.org/10.1007/978-0-8176-4815-2_15
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DOI: https://doi.org/10.1007/978-0-8176-4815-2_15
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