Error bounds for Gauss–Lobatto quadrature of analytic functions on an ellipse

Abstract

For the ($n+2$)-point Gauss–Jacobi–Lobatto quadrature to integrals with the Jacobi weight function ${(1-t)}^{\alpha }{(1+t)}^{\beta }$ ($\alpha >-1$, $\beta >-1$) over the interval $[-1,1]$, we estimate the location where the kernel of the error functional for functions analytic on an ellipse and its interior in the complex plane attains its maximum modulus. As in our previous work on the Gauss–Jacobi rule, when $\alpha \ne \beta$, the location is the intersection point of the ellipse with the real axis in the complex plane. When $\alpha =\beta$ (the Gegenbauer weight), it is the intersection points with the real axis for $-1<\alpha \le 0$ or for $0<\alpha \le \frac{5}{2}$ and $1\le n\le n_{+}\left(\alpha \right)$, and with the imaginary axis for $0<\alpha \le \frac{5}{2}$ and $n>n_{+}\left(\alpha \right)$ or for $\alpha >\frac{5}{2}$. Here, $n_{+}\left(\alpha \right)$ ($>0$) is a monotonously decreasing function for $\alpha >0$ with $n_{+}\left(\frac{5}{2}\right)=1$ and ${lim}_{\alpha \to +0}{n}_{+}\left(\alpha \right)=\infty$. Some numerical results are given to confirm the locations.