On the numerical stability of Fourier extensions
In this excellent paper, an analytic, nonperiodic function \(f:\, [-1,\,1]\to {\mathbb R}\) is approximated by a continuous/discrete Fourier extension (FE) on an extended interval \([-T,\,T]\) with \(T>1\). As shown by \textit{D. Huybrechs} [SIAM J. Numer. Anal. 47, No. 6, 4326--4355 (2010; Zbl 1209.65153)] and by \textit{B. Adcock} and \textit{D. Huybrechs} [J. Comput. Appl. Math. 260, 312--336 (2014; Zbl 1293.65177)], the continuous and discrete FEs converge geometrically fast to \(f\) in the truncation parameter. But computing a FE requires solving an ill-conditioned linear system. Bounds for the corresponding condition numbers are derived. Now the authors show that the FEs are numerically stable when computed in finite arithmetic, and achieve at least superalgebraic convergence rate. Thus, ill-conditioning of the linear system does not prohibit a good approximation of \(f\) by FE. The reason for this disparity is the redundancy of the frame \(\{\exp \frac{i\pi n x}{T}\), \(n\in {\mathbb Z}\}\) in \(L^2(-1,1)\). Finally, the authors consider the computation of discrete FEs from given equispaced data \(\{f(\frac{n}{M})\), \(n=-M,\dots,M\}\). As \textit{R. B. Platte} et al. [SIAM Rev. 53, No. 2, 308--318 (2011; Zbl 1247.41001)] have shown, no method for this problem can be both numerically stable and exponentially convergent. Now the authors prove at least superalgebraic convergence of FE in a numerically stable manner. Numerical examples demonstrate the high performance of these results.
scientific article; zbMATH DE number 6339553
scientific article