  The convergence rates on polynomial interpolation in most cases are estimated
by Lebesgue constants. These estimates may be overestimated for some special
points of sets for functions of limited regularities. In this paper, by
applying the Peano kernel theorem and Wainerman's lemma, new formulas on the
convergence rates are considered. Based upon these new estimates, it shows that
the interpolation at strongly normal pointsystems can achieve the optimal
convergence rate, the same as the best polynomial approximation. Furthermore,
by using the asymptotics on Jacobi polynomials, the convergence rates are
established for Gauss-Jacobi, Jacobi-Gauss-Lobatto or Jacobi-Gauss-Radau
pointsystems. From these results, we see that the interpolations at the
Gauss-Legendre, Legendre-Gauss-Lobatto pointsystem, or at strongly normal
pointsystems, has essentially the same approximation accuracy compared with
those at the two Chebyshev piontsystems, which also illustrates the equally
accuracy of the Gauss and Clenshaw-Curtis quadrature. In addition, numerical
examples illustrate the perfect coincidence with the estimates, which means the
convergence rates are optimal.
