Generalized Bernstein Polynomials and Symmetric Functions

We begin by classifying all solutions of two natural recurrences that Bernstein polynomials satisfy. The first scheme gives a natural characterization of Stancu polynomials. In Section 2, we identify the Bernstein polynomials as coefficients in the generating function for the elementary symmetric functions, which gives a new proof of total positivity for Bernstein polynomials, by identifying the required determinants as Schur functions. In the final section, we introduce a new class of approximation polynomials based on the symplectic Schur functions. These polynomials are shown to agree with the polynomials introduced by Vaughan Jones in his work on subfactors and knots. We show that they have the same fundamental properties as the usual Bernstein polynomials: variation diminishing (whose proof uses symplectic characters), uniform convergence, and conditions for monotone convergence.