On the Optimal Rates of Convergence for Quadratures Derived from Chebyshev Points

Abstract

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, Fejér'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.