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In this paper we provide a reconstruction algorithm for piecewise-smoothfunctions with a-priori known smoothness and number of discontinuities, fromtheir Fourier coefficients, posessing the maximal possible asymptotic rate ofconvergence -- including the positions of the discontinuities and the pointwisevalues 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 the1990s. The key ingredient of the new algorithm is to use a different set ofEckhoff's equations for reconstructing the location of each discontinuity.Instead of consecutive Fourier samples, we propose to use a "decimated" setwhich is evenly spread throughout the spectrum.

Complete Algebraic Reconstruction of Piecewise-Smooth Functions from Fourier Data