标题： 最优多项式预测测度与极值多项式增长 显示英文标题

标题： Optimal Polynomial Prediction Measures and Extremal Polynomial Growth

摘要： 我们证明了如下问题等价：寻找支撑集为复平面上紧集 \(K\) 的测度，使得在外于 \(K\) 的一点处，不超过 \(n\) 次多项式的最小二乘预测方差达到最小。我们将此与寻找在 \(K\) 上模不超过 1 且在此外部点有极值增长的次数不超过 \(n\) 的多项式的问题等价起来。 我们利用这一结果来找到区间 \([-1,1]\) 在纯虚数点处具有极值增长的多项式。 Erdős 在 1947 年研究了关于实多项式在极值增长方面相关的问题。 显示英文摘要

摘要： We show that the problem of finding the measure supported on a compact subset K of the complex plane such that the variance of the least squares predictor by polynomials of degree at most n at a point exterior to K is a minimum, is equivalent to the problem of finding the polynomial of degree at most n, bounded by 1 on K with extremal growth at this external point. We use this to find the polynomials of extremal growth for the interval [-1,1] at a purely imaginary point. The related problem on the extremal growth of real polynomials was studied by Erd\H{o}s in 1947.