Exponentially accurate approximations to periodic Lipschitz functions based on Fourier series partial sums
The purpose of this interesting paper is to present an accurate and robust technique to estimate the locations of discontinuities and the associated jumps in a piecewise smooth, \(2\pi\)-periodic function \(f\), if the first \(2N+ 1\) Fourier coefficients of \(f\) are known. The method consists of two steps: 1. Find initial estimates of the locations and magnitudes of the discontinuities in \(f\) by certain properties of Gibbs phenomenon. 2. Refine these estimates by fitting the asymptotic form of the Fourier coefficients to the given coefficients using a least squares method. Then this procedure, the so-called least squares parameter estimation method, is extended to find discontinuities in \(f^{(k)}\) \((k=1,\dots, M)\). Some aspects of the robustness of this method are discussed. Further, this method is used to reconstruct the original function \(f\), as well as its first few derivatives, with exponential accuracy, if the Fourier coefficients of \(f\) are known. The technique is illustrated with several examples.
scientific article; zbMATH DE number 1346134