Padé-based Interpretation and Correction of the Gibbs Phenomenon

Abstract. The convergence of a Fourier series on an interval can be interpreted naturally as the convergence of a Laurent series on the unit circle in the complex plane. In turn this Laurent series can be interpreted as the sum of an analytic and a co-analytic Taylor series.The Gibbs phenomenon in this context can be seen as an attempt to approximate a logarithmic branch cut with such series. Conversion of a truncated Taylor series to a Padé approximation does a much better job of approximating on most of the unit circle, but a rational function cannot approximate the jump itself. However, one can modify the traditional Padé approximation to include logarithmic singularities. When the jump locations are known exactly, this process appears to converge exponentially to a discontinuous or nonsmooth function throughout the interval . When the jump locations are not known in advance, standard Padé approximation to the derivative of the original series gives poles that approximate jump locations to what is observed to be fourth-order accuracy. All the procedures have analogs in the case of trigonometric interpolation of equispaced data.