Suppose that $K\subset\C$ is compact and that $z_0\in\C\backslash K$ is an external point. An optimal prediction measure for regression by polynomials of degree at most $n,$ is one for which the variance of the prediction at $z_0$ is as small as possible. Hoel and Levine (\cite{HL}) have considered the case of $K=[-1,1]$ and $z_0=x_0\in \R\backslash [-1,1],$ where they show that the support of the optimal measure is the $n+1$ extremme points of the Chebyshev polynomial $T_n(x)$ and characterizing the optimal weights in terms of absolute values of fundamental interpolating Lagrange polynomials. More recently, \cite{BLO} has given the equivalence of the optimal prediction problem with that of finding polynomials of extremal growth. They also study in detail the case of $K=[-1,1]$ and $z_0=ia\in i\R,$ purely imaginary. In this work we generalize the Hoel-Levine formula to the general case when the support of the optimal measure is a finite set and give a formula for the optimal weights in terms of a $\ell_1$ minimization problem.

翻译：设$K\subset\C$为紧集，且$z_0\in\C\backslash K$为外部点。次数不超过$n$的多项式回归中，使$z_0$处预测方差尽可能小的测度称为最优预测测度。Hoel与Levine (\cite{HL}) 考虑了$K=[-1,1]$且$z_0=x_0\in \R\backslash [-1,1]$的情形，他们证明最优测度的支撑集为切比雪夫多项式$T_n(x)$的$n+1$个极值点，并基于拉格朗日插值基多项式的绝对值刻画了最优权重。近期，\cite{BLO} 建立了最优预测问题与极值增长多项式问题的等价性，并详细研究了$K=[-1,1]$且$z_0=ia\in i\R$（纯虚数）的情形。本文将该Hoel-Levine公式推广至最优测度支撑集为有限集的一般情形，并基于$\ell_1$最小化问题给出了最优权重的表达式。