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

Abstract

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.