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On the Numerical Stability of Fourier Extensions

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Abstract

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.

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Notes

  1. The constant of growth was obtained in private communication with A. Kuijlaars. A closed expression (up to several integrals involving the potential function ϕ for the nodes z n ) can be found for c(γ;T). We omit the full argument as it is rather lengthy, but note that it is based on standard results in potential theory. A general reference is [29].

Abbreviations

T :

Extension parameter

N :

Truncation parameter

M, γ :

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

ϕ n (x):

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

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

Exact continuous, discrete and equispaced FEs

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

Numerical continuous, discrete and equispaced FEs

a :

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]

f e(x), f o(x):

Even and odd parts of the function f(x)

g 1(y), g 2(y), g 1,N (y), g 2,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

h i (z), h i,N (z):

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

m(x):

The mapping x↦z

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

κ(F):

Condition number of a mapping F

N 0, N 1, N 2 :

Breakpoints in convergence

{u n ,σ n ,v n }:

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

Φ n :

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}

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

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

a(γ;T):

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

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Acknowledgements

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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Authors and Affiliations

  1. Department of Mathematics, Purdue University, West Lafayette, IN, USA

    Ben Adcock

  2. Department of Computer Science, Katholieke Universiteit Leuven, Leuven, Belgium

    Daan Huybrechs

  3. Department of Applied Mathematics, E.T.S.I.I. Béjar, University of Salamanca, Salamanca, Spain

    Jesús Martín-Vaquero

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  1. Ben Adcock
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Correspondence to Ben Adcock.

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Communicated by Nira Dyn.

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Adcock, B., Huybrechs, D. & Martín-Vaquero, J. On the Numerical Stability of Fourier Extensions. Found Comput Math 14, 635–687 (2014). https://doi.org/10.1007/s10208-013-9158-8

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