Foundations of Computational Mathematics

, Volume 14, Issue 4, pp 635–687 | Cite as

On the Numerical Stability of Fourier Extensions

  • Ben Adcock
  • Daan Huybrechs
  • Jesús Martín-Vaquero


An effective means to approximate an analytic, nonperiodic function on a bounded interval is by using a Fourier series on a larger domain. When constructed appropriately, this so-called Fourier extension is known to converge geometrically fast in the truncation parameter. Unfortunately, computing a Fourier extension requires solving an ill-conditioned linear system, and hence one might expect such rapid convergence to be destroyed when carrying out computations in finite precision. The purpose of this paper is to show that this is not the case. Specifically, we show that Fourier extensions are actually numerically stable when implemented in finite arithmetic, and achieve a convergence rate that is at least superalgebraic. Thus, in this instance, ill-conditioning of the linear system does not prohibit a good approximation.

In the second part of this paper we consider the issue of computing Fourier extensions from equispaced data. A result of Platte et al. (SIAM Rev. 53(2):308–318, 2011) states that no method for this problem can be both numerically stable and exponentially convergent. We explain how Fourier extensions relate to this theoretical barrier, and demonstrate that they are particularly well suited for this problem: namely, they obtain at least superalgebraic convergence in a numerically stable manner.


Fourier series Fourier extension Convergence Stability Equispaced data 



Extension parameter


Truncation parameter

M, γ

Number of equispaced nodes of the equispaced FE, and the oversampling parameter γ=M/N


The exponential \(\frac{1}{\sqrt{2T}} \mathrm{e}^{{\mathrm{i}}\frac{n \pi}{T} x } \)

\(\mathcal {G}_{N} \), \(\mathcal {S}_{N}\), \(\mathcal {C}_{N}\)

Finite-dimensional spaces of exponentials, sines and cosines

FN, \(\tilde{F}_{N}(f)\), FN,M(f)

Exact continuous, discrete and equispaced FEs

GN, \(\tilde{G}_{N}(f)\), GN,M(f)

Numerical continuous, discrete and equispaced FEs


Vector of coefficients of an FE

A, \(\tilde{A}\), \(\bar{A}\)

Matrices of the continuous, discrete and equispaced FE’s

b, \(\tilde{b}\), \(\bar{b}\)

Data vectors for the continuous, discrete and equispaced FEs

x, y, z

Physical domain variable x∈[−1,1], and the mapped variables y∈[c(T),1] and z∈[−1,1]

fe(x), fo(x)

Even and odd parts of the function f(x)

g1(y), g2(y), g1,N(y), g2,N(y)

Images of f e(x) and \(f_{\mathrm{o}}(x) / \sin \frac{\pi}{T} x\) in the y-domain and their polynomial approximations

hi(z), hi,N(z)

Images of g i and g i,N in the z-domain


The mapping xz

c(T), E(T)

FE constants \(\cos \frac{\pi}{T}\) and \(\cot^{2} ( \frac{\pi}{4 T} )\).

\(\mathcal {B}(\rho) \), \(\mathcal {D}(\rho)\)

Bernstein ellipse in the z-domain and its image in the x-domain


Condition number of a mapping F

N0, N1, N2

Breakpoints in convergence


Singular system of A, \(\tilde{A}\) or \(\bar{A}\)


Fourier series corresponding to v n

\(\mathcal {G}_{N,\epsilon} \), \(\mathcal {G}'_{N,\epsilon} \), \(\mathcal {G}_{N,M,\epsilon}\)

The subspace span{Φ n :σ n >ϵ}

HN,ϵ(f), \(\tilde{H}_{N,\epsilon}(f)\), HN,M,ϵ(f)

Truncated SVD FEs corresponding to the continuous, discrete and equispaced cases


Quantity determining the maximal achievable accuracy of the equispaced FE

L2(I), 〈⋅,⋅〉I, ∥⋅∥I

Space of square-integral functions on a domain I and corresponding inner product and norm

〈⋅,⋅〉, ∥⋅∥

Inner product and norm on L2(−1,1)

\(\mathrm {L}^{2}_{w}(I)\), 〈⋅,⋅〉w,I, ∥⋅∥w,I

Space of square integrable functions with respect to a weight function w and corresponding inner product and norm

∥⋅∥∞,I, ∥⋅∥

Uniform norms on an arbitrary domain I and the interval [−1,1] respectively

Mathematics Subject Classification (2010)

42A10 65T40 42C15 



The authors would like to thank John Boyd, Doug Cochran, Laurent Demanet, Anne Gelb, Anders Hansen, Arieh Iserles, Arno Kuijlaars, Mark Lyon, Nilima Nigam, Sheehan Olver, Rodrigo Platte, Jie Shen and Nick Trefethen for useful discussions and comments. They would also like to thank the anonymous referees for their constructive and helpful remarks.


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Copyright information

© SFoCM 2014

Authors and Affiliations

  • Ben Adcock
    • 1
  • Daan Huybrechs
    • 2
  • Jesús Martín-Vaquero
    • 3
  1. 1.Department of MathematicsPurdue UniversityWest LafayetteUSA
  2. 2.Department of Computer ScienceKatholieke Universiteit LeuvenLeuvenBelgium
  3. 3.Department of Applied Mathematics, E.T.S.I.I. BéjarUniversity of SalamancaSalamancaSpain

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