无穷区间上Wiener有理基函数的推广
A generalization of the Wiener rational basis functions on infinite intervals
AI总结：
本文通过将傅里叶级数识别为雅可比多项式/函数的双正交组合，推广了Wiener正交有理基函数到无穷或半无穷区间，并证明了广义基函数具有稀疏微分矩阵，讨论了FFT应用所需的连接问题。
摘要：
我们提出并推导了在无穷或半无穷区间上用于谱展开的正交有理函数基的推广。最初由Wiener提出的原始函数是将傅里叶基映射并加权到无穷区间。通过将傅里叶级数识别为雅可比多项式/函数的双正交组合，我们能够定义广义傅里叶级数，这些级数经过适当映射到整个实轴并加权后，形成了Wiener基函数的推广。已知原始Wiener有理函数继承了对微分的稀疏Galerkin矩阵，并可以利用快速傅里叶变换（FFT）计算模态系数。我们证明了广义基集也具有稀疏微分矩阵，并讨论了连接问题，这是应用FFT所需的理论发展。
英文摘要：
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, appropriately mapped to the whole real line and weighted, form a generalization of 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 modal 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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