# The diagonalisation of the multivariate Bernstein operator

## Shaun Cooper and Shayne Waldron

## Abstract:

Let $B_n$ be the multivariate Bernstein operator of degree $n$ for a

simplex in $Rs$.

In this paper we show that $B_n$ is diagonalisable

with the same eigenvalues as the univariate Bernstein operator, i.e.,

$$lambda_k^{(n)}:={n!over(n-k)!}{1over n^k}, quad k=1,ldots,n,

qquad 1=lambda_1^{(n)}>lambda_2^{(n)}>cdots>lambda_n^{(n)}>0,$$

and we describe the corresponding eigenfuctions and their properties.

Since $B_n$ reproduces only the linear polynomials,

these are the eigenspace for $lambda_1^{(n)}=1$.

For $k>1$, the $lambda_k^{(n)}$--eigenspace consists of polynomials of exact

degree $k$, which are uniquely determined by their leading term.

These are described in terms of the substitution of the barycentric

coordinates (for the underlying simplex)

into {it elementary eigenfunctions}.

It turns out that

there are eigenfunctions of every degree $k$ which are common to each

$B_n$, $nge k$, for sufficiently large $s$.

The {it limiting eigenfunctions} and their connection with

orthogonal polynomials of several variables is also considered.

**Keywords:**
multivariate Bernstein operator,
diagonalisation,
eigenvalues,
eigenfunctions,
total positivity,
Stirling numbers,
Jacobi polynomials

**Math Review Classification:**
Primary 41A10, 15A18, 38B42
; Secondary 33C45, 41A36

**Length:** 28 pages

**Last Updated:** 16 Sept 2002

## Availability: