ON INTERPOLATION APPROXIMATION: CONVERGENCE RATES FOR POLYNOMIAL INTERPOLATION FOR FUNCTIONS OF LIMITED REGULARITY

摘要

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 point systems can achieve the optimal convergence rates, 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 point systems. From these results, we see that the interpolations at the Gauss-Legendre, Legendre-Gauss-Lobatto point systems, or at strongly normal point systems, have essentially the same approximation accuracy compared with those at the two Chebyshev point systems, which also illustrates the equal accuracy of the Gauss and Clenshaw-Curtis quadratures. In addition, numerical examples illustrate the perfect coincidence with the estimates, which means the convergence rates are optimal.