Foundations of Computational Mathematics

, Volume 19, Issue 2, pp 297–331 | Cite as

Stable Extrapolation of Analytic Functions

  • Laurent Demanet
  • Alex TownsendEmail author


This paper examines the problem of extrapolation of an analytic function for \(x > 1\) given \(N+1\) perturbed samples from an equally spaced grid on \([-1,1]\). For a function f on \([-1,1]\) that is analytic in a Bernstein ellipse with parameter \(\rho > 1\), and for a uniform perturbation level \(\varepsilon \) on the function samples, we construct an asymptotically best extrapolant e(x) as a least squares polynomial approximant of degree \(M^*\) determined explicitly. We show that the extrapolant e(x) converges to f(x) pointwise in the interval \(I_\rho \in [1,(\rho +\rho ^{-1})/2)\) as \(\varepsilon \rightarrow 0\), at a rate given by a x-dependent fractional power of \(\varepsilon \). More precisely, for each \(x \in I_{\rho }\) we have
$$\begin{aligned} |f(x) - e(x)| = \mathcal {O}\left( \varepsilon ^{-\log r(x) / \log \rho } \right) , \quad r(x) = \frac{x+\sqrt{x^2-1}}{\rho }, \end{aligned}$$
up to log factors, provided that an oversampling conditioning is satisfied, viz.
$$\begin{aligned} M^* \le \frac{1}{2} \sqrt{N}, \end{aligned}$$
which is known to be needed from approximation theory. In short, extrapolation enjoys a weak form of stability, up to a fraction of the characteristic smoothness length. The number of function samples does not bear on the size of the extrapolation error provided that it obeys the oversampling condition. We also show that one cannot construct an asymptotically more accurate extrapolant from equally spaced samples than e(x), using any other linear or nonlinear procedure. The proofs involve original statements on the stability of polynomial approximation in the Chebyshev basis from equally spaced samples and these are expected to be of independent interest.


Extrapolation Interpolation Chebyshev polynomials Legendre polynomials Approximation theory 

Mathematics Subject Classification

41A10 65D05 



We wish to thank Mohsin Javed for his correspondence regarding the Euler–Maclaurin error formula in [25]. We are also grateful to Ben Adcock for directing us to the literature on stable reconstruction and telling us about [4]. We also thank Matt Li for spotting a handful of typos in an earlier version of the manuscript. We thank the referee for his/her comments and suggestions. The first author is grateful to AFOSR, ONR, NSF, and Total SA for funding. The work of the second author is partially supported by National Science Foundation Grant No. 1645445.

Supplementary material


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Copyright information

© SFoCM 2018

Authors and Affiliations

  1. 1.Department of MathematicsMassachusetts Institute of TechnologyCambridgeUSA
  2. 2.Department of MathematicsCornell UniversityIthacaUSA

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