## Anti-Gaussian quadrature formulae of Chebyshev type

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Sotirios E. Notaris
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## Abstract:

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.## References

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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: notaris@math.uoa.gr
- 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: https://doi.org/10.1090/mcom/3762
- MathSciNet review: 4473104