对从点值样本逼近解析函数的最佳采样率研究
Optimal sampling rates for approximating analytic functions from pointwise samples
AI总结：
本文研究了从等距点样本逼近解析函数的最优采样率，推导了稳定方法的收敛率，并证明了在非等距点分布下采样率的必要性和充分性。
摘要：
我们考虑了从M+1个不同点处的值逼近区间上解析函数的问题。当点等距分布时，最近的结果（称为不可能定理）表明，稳定方法的最佳收敛率是根指数形式的M，并且任何具有更快指数收敛的算法必须在一定速率下指数病态。该结果基于Coppersmith和Rivlin的经典定理，关于在等距网格上受界多项式的最大行为。本文首先将该定理推广到任意点分布。然后提出适用于一般非等距点的不可能定理扩展，并将其应用于相对于（修正）雅可比权重函数等分布的点。这导致了从此类点稳定逼近的必要采样率。我们证明该速率也是充分的，因此精确量化了（除常数外）通过稳定方法从此类节点分布逼近解析函数的精确采样率。数值结果——基于通过经典Remez算法变种计算最大多项式——证实了我们的主要定理。最后，我们讨论了我们的结果对多项式最小二乘逼近的影响。特别是，我们理论上确认了众所周知的启发式方法，即使用次数低于M的多项式进行稳定最小二乘逼近仅当M足够大，使得存在N个节点子集模仿第N组切比雪夫节点的行为时才可能。
英文摘要：
We consider the problem of approximating an analytic function on a compact interval from its values at $M+1$ distinct points. When the points are equispaced, a recent result (the so-called impossibility theorem) has shown that the best possible convergence rate of a stable method is root-exponential in $M$, and that any method with faster exponential convergence must also be exponentially ill-conditioned at a certain rate. This result hinges on a classical theorem of Coppersmith & Rivlin concerning the maximal behaviour of polynomials bounded on an equispaced grid. In this paper, we first generalize this theorem to arbitrary point distributions. We then present an extension of the impossibility theorem valid for general nonequispaced points, and apply it to the case of points that are equidistributed with respect to (modified) Jacobi weight functions. This leads to a necessary sampling rate for stable approximation from such points. We prove that this rate is also sufficient, and therefore exactly quantify (up to constants) the precise sampling rate for approximating analytic functions from such node distributions with stable methods. Numerical results -- based on computing the maximal polynomial via a variant of the classical Remez algorithm -- confirm our main theorems. Finally, we discuss the implications of our results for polynomial least-squares approximations. In particular, we theoretically confirm the well-known heuristic that stable least-squares approximation using polynomials of degree $N < M$ is possible only once $M$ is sufficiently large for there to be a subset of $N$ of the nodes that mimic the behaviour of the $N$th set of Chebyshev nodes.
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