On Interpolation Approximation: Convergence rates for polynomial interpolation for functions of limited regularity

TL;DR

This paper derives new convergence rate estimates for polynomial interpolation at special points, showing optimal rates for functions with limited regularity and comparing different point systems.

Contribution

It introduces new formulas for convergence rates using the Peano kernel theorem and Wainerman's lemma, demonstrating optimal rates at strongly normal pointsystems.

Findings

Interpolation at strongly normal pointsystems achieves optimal convergence rates.

Gauss-Jacobi and Chebyshev pointsystems have comparable approximation accuracy.

Numerical examples confirm the theoretical convergence rate estimates.

Abstract

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.