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：在基于点值的函数近似中，最小二乘法比插值法更具稳定性，但需以增加采样预算为代价。我们证明，只要样本数量 $m$ 与近似空间维数 $n$ 的比值保持恒定，就能在期望的 $L^2$ 意义下实现接近最优的逼近误差。另一方面，当 $m=n$ 时，我们获得一种稳定性因子为 $n$ 量级的插值策略。所提出的采样算法是基于 arXiv:0808.0163 和 arXiv:1508.03261 的贪心方法，具有多项式计算复杂度。