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We continue our investigation of overcoming the 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 first  N  Gegenbauer expansion coefficients, based on the Gegenbauer polynomials      C k μ  ( x ) C_k^\mu (x)       with the weight function      ( 1 −  x 2 ) μ  −  1 / 2 {(1 - {x^2})^{\mu - 1/2}}       for any constant      μ  ≥  0 \mu \geq 0      , of an      L 1 {L_1}       function      f ( x ) f(x)      , we can construct an  exponentially convergent  approximation to the point values of      f ( x ) f(x)       in any subinterval in which the function is analytic. The proof covers the cases of Chebyshev or Legendre partial sums, which are most common in applications.

On the Gibbs phenomenon. IV. Recovering exponential accuracy in a subinterval from a Gegenbauer partial sum of a piecewise analytic function