Accurate Approximation of Functions with Discontinuities, Using Low Order Fourier Coefficients

Abstract : In previous work we introduced a method of using polynomial splines with appropriate discontinuities to approximate a piecewise smooth function f with jump discontinuities of f and f'. The information used is location of discontinuities, and low order, possibly noisy Fourier coefficients. The number of discontinuities was limited to two at most, and the discontinuities needed to lie at meshpoints in a uniform mesh. We showed that the linear operator corresponding to the method is L(sub 2)-bounded with a modest bound, and thus that the method is L(sub 2)-robust in the presence of noise. In the present paper we develop a new method of analysis which enables us to determine operator bounds that are valid for arbitrarily many discontinuities. The new analysis allows discontinuities to be placed arbitrarily. Given a placement, an initially uniform spline mesh of width h must be used such that nearest meshpoints to discontinuities are at least 4h apart (discontinuities then replace these meshpoints); the number of available Fourier coefficients must be at least three times the number of mesh intervals in a period. The previous work was restricted to quadratic splines; the present work includes cubic splines. Much of the analysis uses exact computations with a computer algebra system. We give an example to illustrate the accuracy of the method using noisy Fourier coefficients.