Exact asymptotic for the remainder of Hermite's quadrature formula for function classes \(W^ 2p\) (Q809295)

Consider the Gauss-Jacobi quadrature formula for Chebyshev's weight (\(\sqrt{1-t^ 2})^{-1}\). It has the form \[ \int (\sqrt{1-t^ 2})^{-1}f(t)dt=\frac{\pi}{n}\sum^{n}_{k=1}f(\cos \frac{2k- 1}{2n}\pi)+R_ n(f). \] Denote \(R_ n=\sup \{R_ n(f):f\in W^ 2_ p\}\). An exact asymptotic estimate of \(R_ n\) is given.