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

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- by Akil C. Narayan and Jan S. Hesthaven PDF
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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.## References

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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
- Email: acnaraya@purdue.edu
**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
- Email: Jan.Hesthaven@brown.edu
- Received by editor(s): May 28, 2009
- Received by editor(s) in revised form: April 11, 2010
- Published electronically: December 16, 2010
- © Copyright 2010
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: Math. Comp.
**80**(2011), 1557-1583 - MSC (2010): Primary 65D15, 41A20, 42A10
- DOI: https://doi.org/10.1090/S0025-5718-2010-02437-8
- MathSciNet review: 2785468