Complete Algebraic Reconstruction of Piecewise-Smooth Functions from Fourier Data

Abstract

In this paper we provide a reconstruction algorithm for piecewise-smooth functions with a-priori known smoothness and number of discontinuities, from their Fourier coefficients, posessing the maximal possible asymptotic rate of convergence -- including the positions of the discontinuities and the pointwise values of the function. This algorithm is a modification of our earlier method, which is in turn based on the algebraic method of K.Eckhoff proposed in the 1990s. The key ingredient of the new algorithm is to use a different set of Eckhoff's equations for reconstructing the location of each discontinuity. Instead of consecutive Fourier samples, we propose to use a "decimated" set which is evenly spread throughout the spectrum.

References in corpus (4)

• Signal Acquisition from Measurements via Non-Linear Models
• Reconstruction of Planar Domains from Partial Integral Measurements
• An "algebraic" reconstruction of piecewise-smooth functions from integral measurements
• Decimated generalized Prony systems

Cited by in corpus (7)

• Decimated generalized Prony systems
• Algebraic signal sampling, Gibbs phenomenon and Prony-type systems
• On the norm of inverses of confluent Vandermonde matrices
• Moment vanishing of piecewise solutions of linear ODEs
• Decoupling of Fourier Reconstruction System for Shifts of Several Signals
• Modal occupation measures and LMI relaxations for nonlinear switched systems control
• Sampling, Metric Entropy and Dimensionality Reduction