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.