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

Authors: David Gottlieb and Chi-Wang Shu
Journal: Math. Comp. 64 (1995), 1081-1095
MSC: Primary 42A10; Secondary 33C45, 41A10
MathSciNet review: 1284667
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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 $C_k^\mu (x)$ 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.

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Keywords: Gibbs phenomenon, Gegenbauer polynomials, exponential accuracy
Article copyright: © Copyright 1995 American Mathematical Society