Generalized Bernstein Polynomials and Symmetric Functions
Source: Advances in Applied Mathematics, Volume 28, Number 1, January 2002 , pp. 17-39(23)
Publisher: Academic Press
Abstract: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. © 2002 Elsevier Science (USA).
Document Type: Research Article
Affiliations: 1: Department of Mathematics and Computer Science, Drexel University, Philadelphia, Pennsylvania, 19104 2: Society for Industrial and Applied Mathematics, University City Science Center, Philadelphia, Pennsylvania, 19104
Publication date: January 2002