Jue Wang^1*, Fabian Waleffe²

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¹ Department of Mathematics, Union College, Schenectady, USA.
² Department of Mathematics, University of Wisconsin-Madison, Madison, USA.
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Abstract: 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 PN(α ,β ) ,α ,β > -1 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 N2 appears for -1 <α < 0 ; there are no large negative eigenvalues for collocations at Gauss-Lobatto points. The asymptotic results indicate unstable eigenvalues for α > 1 . 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.

Keywords: Asymptotic Analysis, Spectral Approximations, Jacobi Polynomials, Collocation, Eigenvalues

Cite this paper: Wang, J. and Waleffe, F. (2014) The Asymptotic Eigenvalues of First-Order Spectral Differentiation Matrices. Journal of Applied Mathematics and Physics, 2, 176-188. doi: 10.4236/jamp.2014.25022.

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