  We introduce and develop a theory of orthogonality with respect to Sobolev
inner products on the real line for sequences of functions with a tridiagonal,
skew-Hermitian differentiation matrix. While a theory of such L2-orthogonal
systems is well established, Sobolev orthogonality requires new concepts and
their analysis. We characterise such systems completely as appropriately
weighed Fourier transforms of orthogonal polynomials and present a number of
illustrative examples, inclusive of a Sobolev-orthogonal system whose leading N
coefficients can be computed in $\mathcal{O}(N \log N)$ operations.
