[Paper Review] A stability barrier for reconstructions from Fourier samples

Ben Adcock, Anders C. Hansen+1 more|arXiv (Cornell University)|Oct 29, 2012

Mathematical functions and polynomials44 references4 citations

TL;DR

This paper establishes a fundamental stability barrier for reconstructing analytic, nonperiodic functions from their first $ m $ Fourier coefficients: any stable method can achieve at best root-exponential convergence in $ m $, and exponential convergence implies exponential ill-conditioning. The authors prove this via a novel estimate on the maximal behavior of polynomials with bounded Fourier series, and demonstrate that Fourier extensions provide a stable, superalgebraic alternative that circumvents the barrier by converging to machine precision.

ABSTRACT

We prove that any stable method for resolving the Gibbs phenomenon - that is, recovering high-order accuracy from the first $m$ Fourier coefficients of an analytic and nonperiodic function - can converge at best root-exponentially fast in $m$. Any method with faster convergence must also be unstable, and in particular, exponential convergence implies exponential ill-conditioning. This result is analogous to a recent theorem of Platte, Trefethen & Kuijlaars concerning recovery from pointwise function values on an equispaced $m$-grid. The main step in our proof is an estimate for the maximal behaviour of a polynomial of degree $n$ with bounded $m$-term Fourier series, which is related to a conjecture of Hrycak & Groechenig. In the second part of the paper we discuss the implications of our main theorem to polynomial-based interpolation and least-squares approaches for overcoming the Gibbs phenomenon. Finally, we consider the use of so-called Fourier extensions as an attractive alternative for this problem. We present numerical results demonstrating rapid convergence in a stable manner.

Motivation & Objective

• To establish a fundamental limit on the convergence rate of stable methods for resolving the Gibbs phenomenon from Fourier coefficients.
• To explain why existing methods with geometric convergence are ill-conditioned and why stable methods typically converge slowly.
• To analyze the theoretical limitations of polynomial-based methods such as inverse polynomial reconstruction and least squares.
• To propose and evaluate Fourier extensions as a stable, high-order alternative for reconstructing analytic functions from Fourier data.

Proposed method

• Derives a new estimate for the maximal behavior of a degree-$ n $ polynomial with bounded $ m $-term Fourier series, extending a result of Coppersmith & Rivlin to the Fourier setting.
• Uses this estimate to prove that any method with convergence rate $ \rho^{-m^\tau} $ for $ \rho > 1 $, $ \tau \in (\frac{1}{2}, 1] $, must have ill-conditioning of order $ \rho^{m^{2\tau - 1}} $, establishing the root-exponential barrier.
• Analyzes the inverse polynomial reconstruction method (IPRM) and shows it is exponentially unstable, and that stabilizing it via least squares reduces convergence to root-exponential.
• Proposes Fourier extensions as a stable alternative, where the approximation is constructed by extending the function to a larger interval and computing its Fourier series.
• Employs numerical experiments to compare the polynomial least squares method and Fourier extensions across various test functions with different singularities.
• Uses a parameter $ T $ to control the extension interval length, balancing condition number and convergence rate in the Fourier extension method.

Experimental results

Research questions

• RQ1 What is the best possible convergence rate achievable by a stable method for reconstructing analytic, nonperiodic functions from their first $ m $ Fourier coefficients?
• RQ2 Why do methods with geometric convergence in practice often exhibit severe ill-conditioning?
• RQ3 Can polynomial-based least squares methods achieve faster than root-exponential convergence while remaining stable?
• RQ4 To what extent can Fourier extensions circumvent the stability barrier by converging to machine precision?
• RQ5 How do different function types—e.g., with boundary layers, real or complex singularities—impact the performance of reconstruction methods?

Key findings

• Any stable method for resolving the Gibbs phenomenon from $ m $ Fourier coefficients can converge at best root-exponentially fast in $ m $, i.e., as $ \rho^{-m^{1/2}} $ for some $ \rho > 1 $.
• Exponential convergence in $ m $ necessarily implies exponential ill-conditioning, meaning such methods are numerically unstable.
• The inverse polynomial reconstruction method (IPRM) is shown to be exponentially ill-conditioned, making it impractical for high accuracy.
• Stabilizing IPRM via least squares reduces the convergence rate to root-exponential, matching the theoretical stability barrier.
• Fourier extensions provide superalgebraic (often geometric) convergence down to machine precision, and outperform polynomial least squares for functions with complex singularities or high-frequency oscillations.
• The choice of the extension parameter $ T $ has minimal impact on error in practice, as increased degrees of freedom are balanced by slower convergence per degree.