Algebraic Fourier reconstruction of piecewise smooth functions

Abstract:

Accurate reconstruction of piecewise smooth functions from a finite number of Fourier coefficients is an important problem in various applications. This problem exhibits an inherent inaccuracy, in particular, the Gibbs phenomenon, and it has been intensively investigated during the last few decades. Several nonlinear reconstruction methods have been proposed in the literature, and it is by now well-established that the “classical” convergence order can be completely restored up to the discontinuities. Still, the maximal accuracy of determining the positions of these discontinuities remains an open question.

In this paper we prove that the locations of the jumps (and subsequently the pointwise values of the function) can be reconstructed with at least “half the classical accuracy”. In particular, we develop a constructive approximation procedure which, given the first $k$ Fourier coefficients of a piecewise $C^{2d+1}$ function, recovers the locations of the jumps with accuracy $\sim k^{-\left (d+2\right )}$, and the values of the function between the jumps with accuracy $\sim k^{-\left (d+1\right )}$ (similar estimates are obtained for the associated jump magnitudes). A key ingredient of the algorithm is to start with the case of a single discontinuity, where a modified version of one of the existing algebraic methods (due to K. Eckhoff) may be applied. It turns out that the additional orders of smoothness produce highly correlated error terms in the Fourier coefficients, which eventually cancel out in the corresponding algebraic equations. To handle more than one jump, we apply a localization procedure via a convolution in the Fourier domain, which eventually preserves the accuracy estimates obtained for the single jump. We provide some numerical results which support the theoretical predictions.