Modified anti-Gaussian quadrature formulae of Chebyshev type

Abstract

In a simple way, we prove that there is no positive measure $d\sigma$ on the interval [a, b] such that the corresponding modified anti-Gaussian quadrature formula $\breve{\mathcal G}^{(\gamma )}_{\ell +1}=\breve{\mathcal G}^{(\gamma _\ell )}_{\ell +1}$ is also a Chebyshev quadrature formula. In the special case, this result applies to the corresponding anti-Gaussian quadrature formula $\breve{\mathcal G}_{\ell +1}$. The latter is recently proved in a different way in Notaris [14, Theorem 2.3]. We also show that the only positive and even measure $d\sigma (x)=d\sigma (-x)$ on the symmetric interval $[-a,a]$, for which the modified anti-Gaussian quadrature formula $\breve{\mathcal G}^{(\gamma )}_{\ell +1}=\breve{\mathcal G}^{(\gamma _\ell )}_{\ell +1}$ has the form $\int _{-a}^{a}f(x)\,d\sigma (x)=\frac{\mu _0}{2}[f(-a)+f(a)]+R^{(\gamma _\ell )}_2(f)$ for $\ell =1$ and $\int _{-a}^{a}f(x)\,d\sigma (x)= w_1 f(-a)+w \sum _{k=1}^{\ell -1} f(t_k) +w_1 f(a)+R^{(\gamma _\ell )}_{\ell +1}(f)$ for all $\ell \ge 2$, is the measure $d\sigma (x)=(a^2-x^2)^{-1/2}\,dx$. This result is a generalization of the one for the anti-Gaussian quadrature formula $\breve{\mathcal G}_{\ell +1}$ in Notaris [14, Theorem 3.1] to the modified anti-Gaussian quadrature formulas $\breve{\mathcal G}^{(\gamma )}_{\ell +1}=\breve{\mathcal G}^{(\gamma _\ell )}_{\ell +1}$.