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Mathematics of Computation

Published by the American Mathematical Society, the Mathematics of Computation (MCOM) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.98.

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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
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by David Gottlieb and Chi-Wang Shu PDF
Math. Comp. 64 (1995), 1081-1095 Request permission

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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Additional Information
  • © Copyright 1995 American Mathematical Society
  • Journal: Math. Comp. 64 (1995), 1081-1095
  • MSC: Primary 42A10; Secondary 33C45, 41A10
  • DOI: https://doi.org/10.1090/S0025-5718-1995-1284667-0
  • MathSciNet review: 1284667