# On the Bernstein-Bézier form of Jacobi polynomials on a simplex

## Shayne Waldron

## Abstract:

Here, we give a simple proof of a new representation for orthogonal polynomials over triangular domains which overcomes the need to make symmetry destroying choices to obtain an orthogonal basis for polynomials of fixed degree by employing redundancy. A formula valid for simplices with Jacobi weights is given, and we exhibit its symmetries by using the Bernstein?Bézier form. From it, we obtain the matrix representing the orthogonal projection onto the space of orthogonal polynomials of fixed degree with respect to the Bernstein basis. The entries of this projection matrix are given explicitly by a multivariate analogue of the Click to view the MathML source hypergeometric function. Along the way we show that a polynomial is a Jacobi polynomial if and only if its Bernstein basis coefficients are a Hahn polynomial. We then discuss the application of these results to surface smoothing problems under linear constraints.

**Keywords:**
Bernstein-Bézier form Bernstein-Durrmeyer operator;
Generalised hypergeometric functions; Hahn polynomials;
Jacobi polynomials; Lauricella function; Tight frame; Surface smoothing

**Math Review Classification:**
Primary 33C45, 42C15;
Secondary 41A36, 33C65

**Length:** 14 pages

**Last Updated:** 16 March 2006

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