On the Numerical Stability of Fourier Extensions

• Published in: Foundations of computational mathematics Vol. 14; no. 4; pp. 635 - 687
• Main Authors: Adcock, Ben, Huybrechs, Daan, Martín-Vaquero, Jesús
• Format: Journal Article
• Language: English
• Published: Boston Springer US 01.08.2014; Springer; Springer Nature B.V
• Subjects: Applications of Mathematics; Applied mathematics; Approximation; Computation; Computer Science; Convergence; Economics; Fourier analysis; Fourier series; Fourier transformations; Linear and Multilinear Algebras; Linear systems; Math Applications in Computer Science; Mathematical analysis; Mathematical models; Mathematical research; Mathematics; Mathematics and Statistics; Matrix Theory; Numerical Analysis; Series; Stability; 65T40; 42A10; Equispaced data; Fourier extension; Fourier series; 42C15; Convergence
• ISSN: 1615-3375; 1615-3383
• DOI: 10.1007/s10208-013-9158-8

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.