On the Gibbs Phenomenon V: Recovering Exponential Accuracy from Collocation Point Values of a Piecewise Analytic Function
نویسندگان
چکیده
The paper presents a method to recover exponential accuracy at all points (including at the discontinuities themselves), from the knowledge of an approximation to the interpolation polynomial (or trigonometrical polynomial). We show that if we are given the collocation point values (or' an highly accurate approximation) at the Gauss or Gauss-Lobatto points, we can reconstruct an uniform exponentially convergent approximation to the function f(x) in any sub-interval of analyticity. The proof covers the cases of Fourier, Chebyshev, Legendre, and more general Gegenbauer collocation methods. 1 Research supported by AFOSR grant 93-0090, ARO grant DAAL03-91-G-0123 and DAAH04-94-G-0205, DARPA grant N00014-91-J-4016, NSF grant DMS-9211820, NASA grant NAGl-l145, and NASA contract NAS1-19480 while the authors were in residence at the Institute for Computer Applications in Science and Engineering (ICASE), NASA Langley Research Center, Hampton, VA 23681-0001.
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