Bernstein Constants, Orthogonal Polynomials and Pade Approximation

1913年，S.N.Bernstein证明了用n次多项式逼近|x|的误差，当n乘以n时，当n趋于无穷大时，收敛到一个正常数。我们有了这个未知常量的新表示法，并希望使其完全显式。我们使用的许多方法都是受正交多项式的启发，对于指数权也将被研究。虽然这个问题涉及多项式逼近，但有理逼近和Pade逼近也是研究的重点。最近，Pi和V Buslaev发现了1961年Baker-Gammel-Wills猜想关于Pade逼近的收敛的反例。但对于在整个平面上是完整的或亚纯的函数，这个猜想可能是正确的。此外，我们还将尝试得到单位球上亚纯函数的Nutall-Pommerenke定理的类似结果。

DMS 0400446Doron LubinskyGeorgia TechBernstein Constants, etc.AbstractIn 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 obtainanalogues of the Nuttall-Pommerenke Theorem for functions meromorphic in the unit ball?