The Diagonalisation of the Multivariate Bernstein Operator Original Research Article

Let Bn be the multivariate Bernstein operator of degree n for a simplex in Rs. In this paper, we show that Bn is diagonalisable with the same eigenvalues as the univariate Bernstein operator, i.e.,λ(n)k≔n!(n−k)!1nk, k=1,…,n, 1=λ(n)1>λ(n)2>···>λ(n)n>0 , and we describe the corresponding eigenfuctions and their properties. Since Bn reproduces only the linear polynomials, these are the eigenspace for λ(n)1=1. For k>1, the λ(n)k-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 elementary eigenfunctions. It turns out that there are eigenfunctions of every degree k which are common to each Bn, n⩾k, for sufficiently large s. The limiting eigenfunctions and their connection with orthogonal polynomials of several variables is also considered.