Fast and stable approximation of analytic functions from equispaced samples via polynomial frames

TL;DR

This paper introduces a polynomial frame method for approximating analytic functions from equispaced samples, achieving stable exponential error decay up to a controllable tolerance, overcoming previous impossibility results.

Contribution

It provides a positive, well-conditioned approximation method using polynomial frames and establishes bounds on polynomial behavior, demonstrating near-optimality of the approach.

Findings

Polynomial frame approximation achieves exponential error decay up to a user-controlled tolerance.

Linear oversampling ensures boundedness of polynomials on the interval.

The method is shown to be essentially optimal through an extended impossibility theorem.

Abstract

We consider approximating analytic functions on the interval $[-1,1]$ from their values at a set of $m+1$ equispaced nodes. A result of Platte, Trefethen \& Kuijlaars states that fast and stable approximation from equispaced samples is generally impossible. In particular, any method that converges exponentially fast must also be exponentially ill-conditioned. We prove a positive counterpart to this `impossibility' theorem. Our `possibility' theorem shows that there is a well-conditioned method that provides exponential decay of the error down to a finite, but user-controlled tolerance $\epsilon > 0$, which in practice can be chosen close to machine epsilon. The method is known as \textit{polynomial frame} approximation or \textit{polynomial extensions}. It uses algebraic polynomials of degree $n$ on an extended interval $[-\gamma,\gamma]$, $\gamma > 1$, to construct an approximation on $[-1,1]$ via a SVD-regularized least-squares fit. A key step in the proof of our main theorem is a new result on the maximal behaviour of a polynomial of degree $n$ on $[-1,1]$ that is simultaneously bounded by one at a set of $m+1$ equispaced nodes in $[-1,1]$ and $1/\epsilon$ on the extended interval $[-\gamma,\gamma]$. We show that linear oversampling, i.e., $m = c n \log(1/\epsilon) / \sqrt{\gamma^2-1}$, is sufficient for uniform boundedness of any such polynomial on $[-1,1]$. This result aside, we also prove an extended impossibility theorem, which shows that such a possibility theorem (and consequently the method of polynomial frame approximation) is essentially optimal.