Complete Algebraic Reconstruction of Piecewise-Smooth Functions from Fourier Data
For researchers in signal processing and numerical analysis, this work provides an improved method for reconstructing piecewise-smooth functions from Fourier data, though it is an incremental modification of existing algebraic methods.
This paper presents a reconstruction algorithm for piecewise-smooth functions from Fourier data that achieves the maximal possible asymptotic convergence rate for both discontinuity locations and pointwise values, by using a decimated set of Fourier samples in Eckhoff's equations.
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.