Anti-Gaussian quadrature formulae of Chebyshev type

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.