Reconstruction of a piecewise constant function from noisy Fourier coefficients by Padè method

The problem of reconstructing a piecewise constant function from a finite number of its Fourier coefficients perturbed by noise is considered. A reconstruction method, based on the computation of the Padè approximants to the Z-transform of the sequence of the noisy Fourier coefficients is proposed. The method is based on the remark that the distribution of the poles of the Padè approximants shows, asymptotically, clusters in the complex plane which allow the identification of the discontinuities of the function. It turns out that the Z-transform is a multiple-valued function and the location of the clusters corresponds to the branch points of such a function. By using this property of the Padè poles, a very effective reconstruction method can be developed. Some numerical experiments are presented to show the feasibility of the method.