Superconvergence of Jacobi–Gauss-Type Spectral Interpolation

Abstract

In this paper, we extend the study of superconvergence properties of Chebyshev-Gauss-type spectral interpolation in Zhang (SIAM J Numer Anal 50(5):2966---2985, 2012) to general Jacobi---Gauss-type interpolation. We follow the same principle as in Zhang (SIAM J Numer Anal 50(5):2966---2985, 2012) to identify superconvergence points from interpolating analytic functions, but rigorous error analysis turns out much more involved even for the Legendre case. We address the implication of this study to functions with limited regularity, that is, at superconvergence points of interpolating analytic functions, the leading term of the interpolation error vanishes, but there is no gain in order of convergence, which is in distinctive contrast with analytic functions. We provide a general framework for exponential convergence and superconvergence analysis. We also obtain interpolation error bounds for Jacobi---Gauss-type interpolation, and explicitly characterize the dependence of the underlying parameters and constants, whenever possible. Moreover, we provide illustrative numerical examples to show tightness of the bounds.