Bernstein Constants, Orthogonal Polynomials and Pade Approximation

[摘要]1913年，S.N. Bernstein证明了用至多n次多项式逼近|x|的误差，当乘以n时，当n趋于无穷时收敛为一个正常数。我们有了这个未知常数的新表示，并希望能使它完全明确。我们使用的许多技术都受到正交多项式的启发，对于指数权重的正交多项式也将进行研究。虽然该问题涉及多项式近似，但有理近似和帕德近似也是研究的热点。最近，PI和V Buslaev发现了1961年关于Pade近似收敛性的Baker-Gammel-Wills猜想的反例。但也许这个猜想对整个平面上的整个或亚纯函数是成立的。同时，我们将尝试得到单位球中亚纯函数的nutall - pommerenke定理的类似物。

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?