Mathematics – Classical Analysis and ODEs
Scientific paper
2010-05-11
Mathematics
Classical Analysis and ODEs
Scientific paper
Accurate reconstruction of piecewise-smooth functions from a finite number of Fourier coefficients is an important problem in various applications. The inherent inaccuracy, in particular the Gibbs phenomenon, is being intensively investigated during the last decades. Several nonlinear reconstruction methods have been proposed, 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 $\sim k^{-(d+2)}$, and the values of the function between the jumps with accuracy $\sim k^{-(d+1)}$ (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 a highly correlated error terms in the Fourier coefficients, which eventually cancel out in the corresponding algebraic equations. To handle more than one jump, we propose to apply a localization procedure via a convolution in the Fourier domain.
Batenkov Dmitry
Yomdin Yosef
No associations
LandOfFree
Algebraic Fourier reconstruction of piecewise smooth functions does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with Algebraic Fourier reconstruction of piecewise smooth functions, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Algebraic Fourier reconstruction of piecewise smooth functions will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-385344