On the numerical stability of Fourier extensions
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Publication:404262
DOI10.1007/S10208-013-9158-8zbMATH Open1298.65198arXiv1206.4111OpenAlexW2113499611WikidataQ117717434 ScholiaQ117717434MaRDI QIDQ404262
Author name not available (Why is that?)
Publication date: 4 September 2014
Published in: (Search for Journal in Brave)
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, Trefethen & Kuijlaars 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.
Full work available at URL: https://arxiv.org/abs/1206.4111
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