Resolution properties of the Fourier method for discontinuous waves (Q1892441)

The wave-resolution properties of the Fourier approximations of a wave function with discontinuities are discussed. Resolution properties of the expansion in Gegenbauer polynomials are considered in order to approximate a non-periodic wave function. In recent work by the authors it has been shown that the Fourier coefficients of a discontinuous function contain enough information to reconstruct with exponential accuracy the coefficients of a rapidly converging Gegenbauer expansion. In this paper the truncation error from a finite Fourier series of a wave function with discontinuities is determined by means of the finite Gegenbauer expansion. Numerical calculations are performed to confirm the theory developed in the paper.