The Asymptotic Eigenvalues of First-Order Spectral Differentiation Matrices

We complete and extend the asymptotic analysis of the spectrum of Jacobi Tau approximations that were first considered by Dubiner. The asymptotic formulas for Jacobi polynomials N P ( , ) , , 1 > − α β α β are derived and confirmed by numerical approximations. More accurate results for the slowest decaying mode are obtained. We explain where the large negative eigenvalues come from. Furthermore, we show that a large negative eigenvalue of order NN22 appears for 1 0 − < < α ; there are no large negative eigenvalues for collocations at Gauss-Lobatto points. The asymptotic results indicate unstable eigenvalues for 1 > α . The eigenvalues for Legendre polynomials are directly related to the roots of the spherical Bessel and Hankel functions that are involved in solving Helmholtz equation inspherical coordinates.