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A Family of Orthogonal Rational Functions and Other Orthogonal Systems with a skew-Hermitian Differentiation Matrix
Journal of Fourier Analysis and Applications ( IF 1.2 ) Pub Date : 2020-01-23 , DOI: 10.1007/s00041-019-09718-5
Arieh Iserles , Marcus Webb

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:

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。
更新日期:2020-01-23
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