Iterative methods based on polynomial interpolation filters to detect discontinuities and recover point values from Fourier data

In previous papers, we proposed new filters based on polynomial interpolation to approximate the point values of a piecewise smooth function f on [0,1] from its Fourier coefficients and derived error estimates. We proved that if all the discontinuity points of f are nodes, we can reconstruct point values of f accurately, even close to the discontinuities. We use the new filters to develop iterative methods for detecting the discontinuity points and, therefore, accurately approximate the point values of the function from its Fourier coefficients.