Accurate and Efficient Reconstruction of Discontinuous Functions from Truncated Series Expansions
Knut S. Eckhoff · Mathematics of Computation · 1993
Knowledge of a truncated Fourier series expansion for a discontinuous $2\pi$-periodic function, or a truncated Chebyshev series expansion for a discontinuous nonperiodic function defined on the interval $[-1, 1]$, 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 $(-1, 1)$ 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.