jats:titleAbstract</jats:title>jats:pIn this paper we explore orthogonal systems in jats:inline-formulajats:alternativesjats:tex-math$$\mathrm {L}_2(\mathbb {R})$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">mml:mrowmml:msubmml:miL</mml:mi>mml:mn2</mml:mn></mml:msub>mml:mrowmml:mo(</mml:mo>mml:miR</mml:mi>mml:mo)</mml:mo></mml:mrow></mml:mrow></mml:math></jats:alternatives></jats:inline-formula> 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 jats:inline-formulajats:alternativesjats:tex-math$$\mathrm {L}_2(\mathbb {R})$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">mml:mrowmml:msubmml:miL</mml:mi>mml:mn2</mml:mn></mml:msub>mml:mrowmml:mo(</mml:mo>mml:miR</mml:mi>mml:mo)</mml:mo></mml:mrow></mml:mrow></mml:math></jats:alternatives></jats:inline-formula>, 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.</jats:p>