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.

中文翻译：

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具有斜-Hermitian微分矩阵的正交有理函数族和其他正交系统

在本文中，我们探索\（\ mathrm {L} _2（\ mathbb {R}）\）中的正交系统，该系统产生了偏斜的Hermitian三对角微分矩阵。令人惊讶的是，允许微分矩阵复杂会导致具有有利属性的特定系列的有理正交函数：它们形成\（\ mathrm {L} _2（\ mathbb {R}）\）的正交基础具有一个简单的显式公式作为有理函数，可以容易地操作，并且膨胀系数等于修正函数的经典傅里叶系数，因此可以快速计算。我们表明，该函数族本质上是唯一具有上述形式的微分矩阵的正交基，并且其系数与变量的单调，可微变化一样，等于修正函数的经典傅立叶系数。还讨论了带有偏斜Hermitian，三对角微分矩阵的其他正交基的示例。