The eigenstructure of the Bernstein operator

by Shaun Cooper and Shayne Waldron

Abstract:

The Bernstein operator B_n reproduces the linear polynomials, which are therefore eigenfunctions corresponding to the eigenvalue 1. We determine the rest of the eigenstructure of B_n. Its eigenvalues are

and the corresponding monic eigenfunctions p⁽ⁿ⁾_k are polynomials of degree k, which have k simple zeros in [0, 1]. By using an explicit formula, it is shown that p⁽ⁿ⁾_k converges as n to a polynomial related to a Jacobi polynomial. Similarly, the dual functionals to p⁽ⁿ⁾_k converge as n to measures that we identity. This diagonal form of the Bernstein operator and its limit, the identity (Weierstrass density theorem), is applied to a number of questions. These include the convergence of iterates of the Bernstein operator and why Lagrange interpolation (at n+1 equally spaced points) fails to converge for all continuous functions whilst the Bernstein approximants do. We also give the eigenstructure of the Kantorovich operator. Previously, the only member of the Bernstein family for which the eigenfunctions were known explicitly was the Bernstein-Durrmeyer operator, which is self adjoint.