ON THE GIBBS PHENOMENON .5. RECOVERING EXPONENTIAL ACCURACY FROM COLLOCATION POINT VALUES OF A PIECEWISE ANALYTIC-FUNCTION

This paper presents a method to recover exponential accuracy at all po ints (including at the discontinuities themselves), from the knowledge of an approximation to the interpolation polynomial (or trigonometric al polynomial). We show that if we are given the collocation point val ues (or a highly accurate approximation) at the Gauss or Gauss-Lobatto points, we can reconstruct an uniform exponentially convergent approx imation to the function f(x) in any sub-interval of analyticity. The p roof covers the cases of Fourier, Chebyshev, Legendre, and more genera l Gegenbauer collocation methods. A numerical example is also provided .