On the Gibbs phenomenon. IV. 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 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 $C_k^\mu (x)$ with the weight function ${(1 - {x^2})^{\mu - 1/2}}$ for any constant $\mu \geq 0$, of an ${L_1}$ function $f(x)$, 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 partial sums, which are most common in applications.