ON THE GIBBS PHENOMENON .4. RECOVERING EXPONENTIAL ACCURACY IN A SUBINTERVAL FROM A GEGENBAUER PARTIAL SUM OF A PIECEWISE ANALYTIC-FUNCTION

We continue our investigation of overcoming the Gibbs phenomenon, i.e, , to obtain exponential accuracy at all points (including at the disco ntinuities themselves), from the knowledge of a spectral partial sum o f a discontinuous but piecewise analytic function. We show that if we are given the first N Gegenbauer expansion coefficients, based on the Gegenbauer polynomials C-k(mu)(x) with the weight function (1 - x(2))( mu-1/2) for any constant mu greater than or equal to 0, of an L(1) fun ction f(x), we can construct an exponentially convergent approximation to the point values of f(x) in any subinterval in which the function is analytic. The proof covers the cases of Chebyshev or Legendre parti al sums, which are most common in applications.