The eigenfunctions of the multivariate Bernstein operator
by
Shayne Waldron
University of Auckland
Coauthors: Shaun Cooper

It is shown that B_n the Bernstein operator of degree n for a simplex in R^s is diagonalisable, with eigenvalues

\lambdak(n):=  n! (n-k)!  1 nk ,     k=1, ... , n,        1=\lambda1(n) > \lambda2(n) > ... > \lambdan(n) > 0.

The \lambda_k⁽ⁿ⁾-eigenspace consists of polynomials of exact degree k that are uniquely determined by their leading term (i.e., the eigenspace is isomorphic to the space of homogeneous polynomials of degree k). These eigenspaces are described in terms of the barycentric coordinates (for the underlying simplex) and their substitution into elementary eigenfunctions. In contrast to the univariate case there are eigenfunctions of every degree k which are common to each B_n, n >= k for sufficiently large s. Time permitting we will discuss the limiting eigenfunctions and their connection with orthogonal polynomials of several variables, and the connection of the diagonal form of B_n with the failure of Lagrange interpolation of degree n at the simplex points to converge for all continuous functions.

Date received: February 10, 1999

Copyright © 1999 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cabp-39.