Optimal sampling rates for approximating analytic functions from pointwise samples

Abstract

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.