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 disco ntinuities themselves, from the knowledge of a spectral partial sum of a discontinuous but piecewise analytic function. We show that if we a re given the first N expansion coefficients of an L(2) function f(x) i n terms of either the trigonometric polynomials or the Legendre polyno mials, we can construct an exponentially convergent approximation to t he point values of f(x) in any sub-interval in which it is analytic.