NDLI: On the Numerical Stability of Fourier Extensions

Content Provider	Springer Nature Link
Author	Martín Vaquero, Jesús Adcock, Ben Huybrechs, Daan
Copyright Year	2014
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.
Ending Page	687
Page Count	53
Starting Page	635
File Format	PDF
ISSN	16153375
e-ISSN	16153383
Journal	Foundations of Computational Mathematics
Issue Number	4
Volume Number	14
Language	English
Publisher	Springer US
Publisher Date	2014-04-15
Publisher Place	Boston
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Stability Economics general Trigonometric approximation and interpolation Linear and Multilinear Algebras, Matrix Theory Fourier series Trigonometric approximation Convergence Numerical Analysis Computer Science Equispaced data Fourier extension General harmonic expansions, frames Applications of Mathematics Math Applications in Computer Science
Content Type	Text
Resource Type	Article
Subject	Applied Mathematics Analysis Computational Theory and Mathematics Computational Mathematics

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