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Orthonormal rational functions on a semi-infinite interval

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  • Liu, Jianqiang

Abstract

In this paper we propose a novel family of weighted orthonormal rational functions on a semi-infinite interval. We write a sequence of integer-coefficient polynomials in several forms and derive their corresponding differential equations. These equations do not form Sturm-Liouville problems. We overcome this disadvantage by multiplying some factors, resulting in a sequence of irrational functions. We deduce various generating functions of this sequence and find the associated Sturm-Liouville problems, which bring orthogonality. Then we establish a Hilbert space of functions defined on a semi-infinite interval with an inner product induced by a weight function determined by the Sturm-Liouville problems mentioned above. The even subsequence of the irrational function sequence above forms an orthonormal basis for this space. The Fourier expansions of some power functions are deduced. The sequence of non-positive integer power functions forms a non-orthogonal basis for this Hilbert space. We give two examples, one example of Fourier expansion and one example of interpolation as applications.

Suggested Citation

  • Liu, Jianqiang, 2024. "Orthonormal rational functions on a semi-infinite interval," Applied Mathematics and Computation, Elsevier, vol. 479(C).
    • Handle: RePEc:eee:apmaco:v:479:y:2024:i:c:s0096300324003485
    DOI: 10.1016/j.amc.2024.128887
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    References listed on IDEAS

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    1. Pucanović, Zoran & Pešović, Marko, 2023. "Chebyshev polynomials and r-circulant matrices," Applied Mathematics and Computation, Elsevier, vol. 437(C).
    2. Zhang, Shangyuan & Nie, Yufeng, 2023. "Localized Chebyshev and MLS collocation methods for solving 2D steady state nonlocal diffusion and peridynamic equations," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 206(C), pages 264-285.
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