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A generalization of the Wiener rational basis functions on infinite intervals: Part I–derivation and properties

Authors: Akil C. Narayan and Jan S. Hesthaven
Journal: Math. Comp. 80 (2011), 1557-1583
MSC (2010): Primary 65D15, 41A20, 42A10
Published electronically: December 16, 2010
MathSciNet review: 2785468
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Abstract: We formulate and derive a generalization of an orthogonal rational-function basis for spectral expansions over the infinite or semi-infinite interval. The original functions, first presented by Wiener, are a mapping and weighting of the Fourier basis to the infinite interval. By identifying the Fourier series as a biorthogonal composition of Jacobi polynomials/functions, we are able to define generalized Fourier series which, when appropriately mapped to the whole real line and weighted, generalize Wiener’s basis functions. It is known that the original Wiener rational functions inherit sparse Galerkin matrices for differentiation, and can utilize the fast Fourier transform (FFT) for computation of the expansion coefficients. We show that the generalized basis sets also have a sparse differentiation matrix and we discuss connection problems, which are necessary theoretical developments for application of the FFT.

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Additional Information

Akil C. Narayan
Affiliation: Department of Mathematics, Purdue University, 150 N. University Street, West Lafayette, Indiana 47907
MR Author ID: 932761

Jan S. Hesthaven
Affiliation: Division of Applied Mathematics, Brown University, 182 George Street, Box F, Providence, Rhode Island 02912
MR Author ID: 350602
ORCID: 0000-0001-8074-1586

Keywords: Spectral methods, infinite interval, rational functions
Received by editor(s): May 28, 2009
Received by editor(s) in revised form: April 11, 2010
Published electronically: December 16, 2010
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.