  In this paper we explore orthogonal systems in $\mathrm{L}_2(\mathbb{R})$
which give rise to a skew-Hermitian, tridiagonal differentiation matrix.
Surprisingly, allowing the differentiation matrix to be complex leads to a
particular family of rational orthogonal functions with favourable properties:
they form an orthonormal basis for $\mathrm{L}_2(\mathbb{R})$, have a simple
explicit formulae as rational functions, can be manipulated easily and the
expansion coefficients are equal to classical Fourier coefficients of a
modified function, hence can be calculated rapidly. We show that this family of
functions is essentially the only orthonormal basis possessing a
differentiation matrix of the above form and whose coefficients are equal to
classical Fourier coefficients of a modified function though a monotone,
differentiable change of variables. Examples of other orthogonal bases with
skew-Hermitian, tridiagonal differentiation matrices are discussed as well.
