Bernstein Constants, Orthogonal Polynomials and Pade Approximation

Georgia Tech Research Corporation, Atlanta GA

Investigators

Abstract

DMS 0400446 Doron Lubinsky Georgia Tech Bernstein Constants, etc. Abstract In 1913, S.N. Bernstein proved that the error in approximation of |x| by polynomials of degree at most n, when multiplied by n, converges to a positive constant as n approaches infinity. We have a new representation of this unknown constant, and hope to make it fully explicit. Many of the techniques we use are inspired by orthogonal polynomials, which for exponential weights will also be investigated. While this problem involves polynomial approximation, rational and Pade approximation is also a focus of the research. Recently the PI and V Buslaev found counterexamples to the 1961 Baker-Gammel-Wills Conjecture on convergence of Pade approximations. But perhaps the conjecture is true for functions that are entire or meromorphic in the whole plane. Also, we will try to obtain analogues of the Nuttall-Pommerenke Theorem for functions meromorphic in the unit ball?