Accurate and efficient reconstruction of discontinuous functions from truncated series expansions

Abstract

Knowledge of a truncated Fourier series expansion for a discontinuous 2 π 2\pi -periodic function, or a truncated Chebyshev series expansion for a discontinuous nonperiodic function defined on the interval [ − 1 , 1 ] [-1, 1] , is used in this paper to accurately and efficiently reconstruct the corresponding discontinuous function. First an algebraic equation of degree M for the M locations of discontinuities in each period for a periodic function, or in the interval ( − 1 , 1 ) (-1, 1) for a nonperiodic function, is constructed. The M coefficients in that algebraic equation of degree M are obtained by solving a linear algebraic system of equations determined by the coefficients in the known truncated expansion. By solving an additional linear algebraic system for the M jumps of the function at the calculated discontinuity locations, we are able to reconstruct the discontinuous function as a linear combination of step functions and a continuous function.