Stable Extrapolation of Analytic Functions



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.


  1. 1.
    B. Adcock and A. C. Hansen, Stable reconstructions in Hilbert spaces and the resolution of the Gibbs phenomenon, Appl. Comput. Harm. Anal., 32 (2012), pp. 357–388.MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    B. Adcock and A. C. Hansen, Generalized sampling and the stable and accurate reconstruction of piecewise analytic functions from their Fourier coefficients, Math. Comput., 84 (2015), pp. 237–270.MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    B. Adcock, A. C. Hansen, and A. Shadrin, A stability barrier for reconstructions from Fourier samples, SIAM J. Numer. Anal., 52 (2014), pp. 125–139.MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    B. Adcock and R. Platte, A mapped polynomial method for high-accuracy approximations on arbitrary grids, SIAM J. Numer. Anal., 54 (2016), pp. 2256–2281MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    B. K. Alpert and V. Rokhlin, A fast algorithm for the evaluation of Legendre expansions, SIAM J. Sci. Stat. Comput., 12 (1991), pp. 158–179.MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    P. Borwein and T. Erdelyi, Polynomials and Polynomial Inequalities, Graduate Texts in Mathematics, , Springer, 1995.Google Scholar
  7. 7.
    J. P. Boyd, Defeating the Runge phenomenon for equally spaced polynomial interpolation via Tikhonov regularization, Appl. Math. Letters, 5 (1992), pp. 57–59.MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    J. P. Boyd and J. R. Ong, Exponentially-convergent strategies for defeating the Runge phenomenon for the approximation of non-periodic functions, Part I: Single-interval schemes, Commun. Comput. Phys., 5 (2009), pp. 484–497.MathSciNetMATHGoogle Scholar
  9. 9.
    L. Brutman, Lebesgue functions for polynomial interpolation—a survey, Annals of Numer. Math., 4 (1996), pp. 111–128.MathSciNetMATHGoogle Scholar
  10. 10.
    E. J. Candès, J. Romberg, and T. Tao, Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information, IEEE Trans. Inform. Theory 52 (2006) pp. 489–509.MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    E. J. Candès and C. Fernandez-Granda, Towards a mathematical theory of super-resolution, Communications on Pure and Applied Mathematics, 67 (2014) pp. 906–956.MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    A. Cohen, M. A. Davenport, and D. Leviatan, On the stability and accuracy of least squares approximations, Found. Comput. Math., 13 (2013), pp. 819–834.MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    G. Dahlquist and Å. Björck, Numerical Methods, Dover edition, unbridged republication of Prentice-Hall, 2003.Google Scholar
  14. 14.
    L. Demanet, M. Ferrara, N. Maxwell, J. Poulson, and L. Ying, A butterfly algorithm for synthetic aperture radar imaging, SIAM J. Imag. Sci. 5-1 (2012) pp. 203–243MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    L. Demanet, D. Needell, and N. Nguyen, Super-resolution via superset selection and pruning, arXiv preprint arXiv:1302.6288, (2013).
  16. 16.
    L. Demanet and N. Nguyen, The recoverability limit for superresolution via sparsity, arXiv preprint arXiv:1502.01385, (2015).
  17. 17.
    L. Demanet and L. Ying, On Chebyshev interpolation of analytic functions, MIT technical report, 2010Google Scholar
  18. 18.
    T. A. Driscoll, N. Hale, and L. N. Trefethen, editors, Chebfun Guide, Pafnuty Publications, Oxford, 2014.Google Scholar
  19. 19.
    W. Gautschi, How (un)stable are Vandermonde systems, Asymp. Comput. Anal., 124 (1990), pp. 193–210.MathSciNetMATHGoogle Scholar
  20. 20.
    W. Gautschi, Optimally scaled and optimally conditioned Vandermonde and Vandermonde-like matrices, BIT Numer. Math., 51 (2011), pp. 103–125.MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    M. Goldberg, E. Tadmor, and G. Zwas, Numerical radius of positive matrices, Linear Alg. Appl., 12 (1975), pp. 209–214.MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    G. Golub and C. Van Loan, Matrix Computations, John Hopkins, Baltimore, 1996.MATHGoogle Scholar
  23. 23.
    N. Hale and A. Townsend, A fast, simple, and stable Chebyshev–Legendre transform using an asymptotic formula, SIAM J. Sci. Comput., 36 (2014), A148–A167.MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    N. Hale and A. Townsend, A fast FFT-based discrete Legendre transform, IMA J. Numer. Anal., 36 (2015), pp. 1670–1684.MathSciNetCrossRefGoogle Scholar
  25. 25.
    M. Javed and L. N. Trefethen, A trapezoidal rule error bound unifying the Euler–Maclaurin formula and geometric convergence for periodic functions, Proc. Roy. Soc. London A, 470 (2014).Google Scholar
  26. 26.
    F. W. J. Olver, D. W. Lozier, R. F. Boisvert, and C. Clark, NIST Handbook of Mathematical Functions, Cambridge University Press, 2010.Google Scholar
  27. 27.
    R. B. Platte, L. N. Trefethen, and A. B. J. Kuijlaars, Impossibility of fast stable approximation of analytic functions from equally spaced samples, SIAM Review, 53 (2011), pp. 308–318.MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    M. J. D. Powell, Approximation Theory and Methods, Cambridge University Press, 1981.Google Scholar
  29. 29.
    L. Reichel and G. Opfer, Chebyshev-Vandermonde systems, Math. Comput., 57 (1991), pp. 703–721.MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    C. Runge, Über empirische Funktionen und die Interpolation zwischen äquidistanten Ordinaten, Zeitschrift für Mathematik und Physik, 46 (1901), pp. 224–243.MATHGoogle Scholar
  31. 31.
    S. Schechter, On the inversion of certain matrices, Mathematical Tables and Other Aids to Computation, 13 (1959), pp. 73–77.MathSciNetCrossRefMATHGoogle Scholar
  32. 32.
    G. Strang, The discrete cosine transform, SIAM Review, 41 (1999), pp. 135–147.MathSciNetCrossRefMATHGoogle Scholar
  33. 33.
    L. N. Trefethen and J. A. C. Weideman, Two results on polynomial interpolation in equally spaced points, J. Approx. Theory, 65 (1991), pp. 247–260.MathSciNetCrossRefMATHGoogle Scholar
  34. 34.
    L. N. Trefethen, Approximation Theory and Approximation Practice, SIAM, 2013.Google Scholar
  35. 35.
    J. G. Wendel, Note on the gamma function, Amer. Math. Monthly, 55 (1948), pp. 563–564.MathSciNetCrossRefGoogle Scholar
  36. 36.
    H. Weyl, Das asymptotische Verteilungsgesetz der Eigenwerte linearer partieller Differentialgleichungen, Mathematische Annalen, 71 (1912), pp. 441–479.MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© SFoCM 2018

Authors and Affiliations

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

Personalised recommendations