On the Gibbs phenomenon .3. Recovering exponential accuracy in a sub-interval from a spectral partial sum of a piecewise 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 L(2) 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.