Exponentially Accurate Approximations to Periodic Lipschitz Functions Based on Fourier Series Partial Sums

The problem of accurately reconstructing a piecewise smooth. 2π-periodic function f and its first few derivatives, given only a truncated Fourier series representation of f, is studied and solved. The reconstruction process is divided into two steps. In the first step, the first 2N + 1 Fourier coefficients of f are used to approximate the locations and magnitudes of the discontinuities in f and its first M derivatives. This is accomplished by first finding initial estimates of these quantities based on certain properties of Gibbs phenomenon, and then refining these estimates by fitting the asymptotic form of the Fourier coefficients to the given coefficients using a least-squares approach. The locations of the singularities are approximated to within O(N−M−2), and the associated jump of the kth derivative of f is approximated to within O(N−M−1+k), as N → ∞, and the method is robust. These estimates are then used with a class of singular basis functions, which have certain “built-in” singularities, to construct a new sequence of approximations to f. Each of these new approximations is the sum of a piecewise smooth function and a new Fourier series partial sum. When N is proportional to M, it is shown that these new approximations, and their derivatives, converge exponentially in the maximum norm to f, and its corresponding derivatives, except in the union of a finite number of small open intervals containing the points of singularity of f. The total measure of these intervals decreases exponentially to zero as M → ∞. The technique is illustrated with several examples.