On the Gibbs Phenomenon IV: Recovering Exponential Accuracy in a Sub- Interval From a Gegenbauer Partial Sum of a Piecewise Analytic Function
INSTITUTE FOR COMPUTER APPLICATIONS IN SCIENCE AND ENGINEERING HAMPTON VA
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We continue our investigation of overcoming 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 with the weight function of an L function fx, we can construct an exponentially convergent approximation to the point values of fx in any sub-interval in which the function is analytic. The proof covers the cases of Chebyshev or Legendre partial sums, which are most common in applications.
- Numerical Mathematics