On the Gibbs Phenomenon Iv: Recovering Exponential Accuracy in a Sub-interval from a Gegenbauer Partial Sum of a Piecewise Analytic Function

We continue our investigation of overcoming Gibbs phenomenon, i.e., to obtain 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 rst N Gegenbauer expansion coe cients, based on the Gegenbauer polynomials C k (x) with the weight function (1 x ) 1 2 for any constant 0, of an L1 function f(x), we can construct an exponentially convergent approximation to the point values of f(x) in any sub-interval in which the function is analytic. The proof covers the cases of Chebyshev or Legendre partial sums, which are most common in applications. Research supported by AFOSR grant 93-0090, ARO grant DAAL03-91-G-0123, DARPA grant N0001491-J-4016, NSF grant DMS-9211820, NASA grant NAG1-1145, and NASA contract NAS1-19480 while the authors were in residence at ICASE, NASA Langley Research Center, Hampton, VA 23665.