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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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Anti-Gaussian quadrature formulae of Chebyshev type
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by Sotirios E. Notaris HTML | PDF
Math. Comp. 91 (2022), 2803-2816 Request permission


We prove that there is no positive measure $d\sigma$ on the interval $[a,b]$ such that the corresponding anti-Gaussian quadrature formula is also a Chebyshev quadrature formula. We also show that the only positive and even measure $d\sigma (t)=d\sigma (-t)$ on the symmetric interval $[-a,a]$, for which the anti-Gaussian formula has the form $\int _{-a}^{a}f(t)d\sigma (t)=\frac {\mu _{0}}{2}[f(a)+f(-a)]+R_{2}^{AG}(f)$ for $n=1$ and $\int _{-a}^{a}f(t)d\sigma (t)=w_{1}f(a)+w\sum _{\mu =2}^{n}f(t_{\mu })+w_{1}f(-a)+R_{n+1}^{AG}(f)$ for all $n\geq 2$, is the measure $d\sigma (t)=(a^{2}-t^{2})^{-1/2}dt$. It turns out that the formula for $n\geq 2$ is the $(n-1)$-point Gauss-Lobatto quadrature formula for the measure $d\sigma (t)=(a^{2}-t^{2})^{-1/2}dt$, which is a generalization of what happens in the case of the Chebyshev measure of the first kind. Moreover, we compute the anti-Gaussian formulae for any one of the four Chebyshev measures.
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Additional Information
  • Sotirios E. Notaris
  • Affiliation: Department of Mathematics, National and Kapodistrian University of Athens, Pane-pistemiopolis, 15784 Athens, Greece
  • ORCID: 0000-0002-6542-5021
  • Email:
  • Received by editor(s): August 15, 2021
  • Received by editor(s) in revised form: March 25, 2022
  • Published electronically: August 3, 2022
  • Additional Notes: Dedicated to the memory of Professor Dirk P. Laurie (1946-2019).
  • © Copyright 2022 American Mathematical Society
  • Journal: Math. Comp. 91 (2022), 2803-2816
  • MSC (2020): Primary 41A55, 33C45; Secondary 65D32
  • DOI:
  • MathSciNet review: 4473104