In approximation of functions based on point values, least-squares methods provide more stability than interpolation, at the expense of increasing the sampling budget. We show that near-optimal approximation error can nevertheless be achieved, in an expected $L^2$ sense, as soon as the sample size $m$ is larger than the dimension $n$ of the approximation space by a constant ratio. On the other hand, for $m=n$, we obtain an interpolation strategy with a stability factor of order $n$. The proposed sampling algorithms are greedy procedures based on arXiv:0808.0163 and arXiv:1508.03261, with polynomial computational complexity.

翻译：在基于点值的函数逼近中，最小二乘法相比插值法具有更强的稳定性，但代价是增加了采样预算。我们证明，只要样本量 $m$ 与逼近空间维数 $n$ 之比为常数，即可在期望的 $L^2$ 意义下达到近最优逼近误差。另一方面，当 $m=n$ 时，我们得到一种稳定因子为 $n$ 阶的插值策略。所提出的采样算法是基于 arXiv:0808.0163 和 arXiv:1508.03261 的贪心过程，具有多项式计算复杂度。