通过 $\ell_1$ 最小化刻画最优预测测度
A Characterization of Optimal Prediction Measures via $\ell_1$ Minimization
AI总结：
本文研究紧集外点处多项式回归的最优预测测度，将 Hoel–Levine 权重公式推广到有限支集情形，并用 $\ell_1$ 最小化刻画最优权重。
摘要：
假设 $K\subset\C$ 是紧集，且 $z_0\in\C\backslash K$ 是一个外部点。对于次数至多为 $n$ 的多项式回归而言，最优预测测度是指使在 $z_0$ 处预测方差尽可能小的测度。Hoel 和 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集是 Chebyshev 多项式 $T_n(x)$ 的 $n+1$ 个极值点，并用基本插值 Lagrange 多项式的绝对值刻画了最优权重。最近，\cite{BLO} 给出了最优预测问题与寻找极端增长多项式问题之间的等价性。他们还详细研究了 $K=[-1,1]$ 且 $z_0=ia\in i\R$（即纯虚数）的情形。在本工作中，我们将 Hoel–Levine 公式推广到最优测度支集为有限集的一般情形，并依据一个 $\ell_1$ 最小化问题给出最优权重的公式。
英文摘要：
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.
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