On the Gibbs Phenomenon III: Recovering Exponential Accuracy in a Sub-Interval From a Spectral Partial Sum of a Pecewise Analytic Function

We continue the investigation of overcoming the Gibbs phenomenon, i.e., obtaining exponential accuracy at all points, including at the discontinuities themselves, from the knowledge of a spectral partial sum of a discontinuous but piecewise analytic function. We show that if we are given the first N expansion coefficients of an L² function $f(x)$ in terms of either the trigonometric polynomials or the Legendre polynomials, we can construct an exponentially convergent approximation to the point values of $f(x)$ in any sub-interval in which it is analytic.