On the numerical stability of Fourier extensions
type: Item
instance of: scholarly article
title: On the numerical stability of Fourier extensions
zbMATH Open document ID: 1298.65198
author: Ben Adcock
author: Daan Huybrechs
author: J. Martín-Vaquero
published in: Foundations of Computational Mathematics
publication date: 2014-09-04T00:00:00Z
full work available at URL: https://arxiv.org/abs/1206.4111
full work available at URL: https://lirias.kuleuven.be/handle/123456789/351524
review text: 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.
reviewed by: Manfred Tasche
Mathematics Subject Classification ID: 65T40
Mathematics Subject Classification ID: 42A10
Mathematics Subject Classification ID: 42C15
Mathematics Subject Classification ID: 65F22
zbMATH DE Number: 6339553
zbMATH Keywords: Fourier extension
zbMATH Keywords: nonperiodic function
zbMATH Keywords: numerical stability
zbMATH Keywords: frame approximation
zbMATH Keywords: convergence rate
zbMATH Keywords: condition number
zbMATH Keywords: numerical Fourier extension
zbMATH Keywords: equispaced data
zbMATH Keywords: Runge phenomenon
zbMATH Keywords: numerical examples
MaRDI profile type: Publication
OpenAlex ID: W2113499611
Wikidata QID: Q117717434
arXiv ID: 1206.4111
cites work: On the resolution power of Fourier extensions for oscillatory functions
cites work: A spectral FC solver for the compressible Navier-Stokes equations in general domains. I: Explicit time-stepping
cites work: Q5820721
cites work: Q3993205
cites work: A comparison of numerical algorithms for Fourier extension of the first, second, and third kinds
cites work: Fourier embedded domain methods: Extending a function defined on an irregular region to a rectangle so that the extension is spatially periodic and \(C^{\infty}\)
cites work: Trouble with Gegenbauer reconstruction for defeating Gibbs' phenomenon: Runge phenomenon in the diagonal limit of Gegenbauer polynomial approximations
cites work: Q5268582
cites work: Divergence (Runge phenomenon) for least-squares polynomial approximation on an equispaced grid and mock-Chebyshev subset interpolation
cites work: Q4450071
cites work: High-order unconditionally stable FC-AD solvers for general smooth domains. I: Basic elements
cites work: Accurate, high-order representation of complex three-dimensional surfaces via Fourier continuation analysis
cites work: Spectral Methods
cites work: An introduction to frames and Riesz bases
cites work: The Growth of Polynomials Bounded at Equally Spaced Points
cites work: A Class of Nonharmonic Fourier Series
cites work: The Future Fast Fourier Transform?
cites work: A Practical Guide to Pseudospectral Methods
cites work: Q4199478
cites work: On the Gibbs Phenomenon and Its Resolution
cites work: On the Gibbs phenomenon. I: Recovering exponential accuracy from the Fourier partial sum of a nonperiodic analytic function
cites work: On the Fourier Extension of Nonperiodic Functions
cites work: A modified Chebyshev pseudospectral method with an \(O(N^{-1})\) time step restriction
cites work: Approximation error in regularized SVD-based Fourier continuations
cites work: A Fast Algorithm for Fourier Continuation
cites work: High-order unconditionally stable FC-AD solvers for general smooth domains. II: Elliptic, parabolic and hyperbolic PDEs; theoretical considerations
cites work: A spectral embedding method applied to the advection-diffusion equation
cites work: Impossibility of Fast Stable Approximation of Analytic Functions from Equispaced Samples
cites work: Q4328333
cites work: Q3997537
cites work: Prolate Spheroidal Wave Functions, Fourier Analysis, and Uncertainty-V: The Discrete Case
cites work: Q4337625
cites work: The prolate matrix
DOI: 10.1007/S10208-013-9158-8
description: scientific article; zbMATH DE number 6339553