A generalization of the Wiener rational basis functions on infinite intervals

TL;DR

This paper extends Wiener rational basis functions to a more general form suitable for spectral expansions on infinite intervals, maintaining computational efficiency and sparse differentiation matrices.

Contribution

It introduces a generalized class of Wiener rational basis functions derived via Jacobi polynomials, enabling spectral methods on infinite domains with efficient computation.

Findings

Generalized basis functions have sparse differentiation matrices.

The new basis maintains compatibility with FFT-based computations.

Theoretical development of connection problems for the generalized basis.

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, 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.