  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.
