  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.
