The diagonalisation of the multivariate Bernstein operator

Shaun Cooper and Shayne Waldron


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