Approximation and interpolation with Bernstein polynomials.

Abstract

Bernstein polynomials, long a staple of approximation theory and computational geometry, have also increasingly become of interest in finite element methods. In this dissertation, we investigate fundamental problems in approximation theory and numerical analysis involving Bernstein polynomials. We begin by developing a structured decomposition of the inverse of the matrices related to approximation and interpolation. These matrices are highly ill-conditioned, and so we introduce a nonstandard matrix norm to study the conditioning of the matrices, showing that the conditioning in this case is better than in the standard 2-norm. We conclude by giving an algorithm for enforcing bounds constraints on the approximating polynomial. Extensions of the interpolation problem and constrained approximation problem to higher dimensions are also considered.