  In this paper, we study the optimal general convergence rates for quadratures
derived from Chebyshev points. By building on the aliasing errors on
integration of Chebyshev polynomials, together with the asymptotic formulae on
the coefficients of Chebyshev expansions, new and optimal convergence rates for
$n$-point Clenshaw-Curtis, Fej\'{e}r's first and second quadrature rules are
established for Jacobi weights or Jacobi weights multiplied by $\ln((x+1)/2)$.
The convergence orders are attainable for some functions of finite
regularities. In addition, by using refined estimates on aliasing errors on
integration of Chebyshev polynomials by Gauss-Legendre quadrature, an improved
convergence rate for Gauss-Legendre is given too.
