Orthogonal systems in L₂(ℝ), once implemented in spectral methods, enjoy a number of important advantages if their differentiation matrix is skew‐symmetric and highly structured. Such systems, where the differentiation matrix is skew‐symmetric, tridiagonal, and irreducible, have been recently fully characterised. In this paper we go a step further, imposing the extra requirement of fast computation: specifically, that the first N coefficients of the expansion can be computed to high accuracy in operations. We consider two settings, one approximating a function f directly in (−∞, ∞) and the other approximating [f(x) + f(−x)]/2 and [f(x) − f(−x)]/2 separately in [0, ∞). In each setting we prove that there is a single family, parametrised by α, β > − 1, of orthogonal systems with a skew‐symmetric, tridiagonal, irreducible differentiation matrix and whose coefficients can be computed as Jacobi polynomial coefficients of a modified function. The four special cases where α, β = ± 1/2 are of particular interest, since coefficients can be computed using fast sine and cosine transforms. Banded, Toeplitz‐plus‐Hankel multiplication operators are also possible for representing variable coefficients in a spectral method. In Fourier space these orthogonal systems are related to an apparently new generalisation of the Carlitz polynomials. © 2020 Wiley Periodicals, Inc.

中文翻译：

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具有偏对称微分矩阵的正交系统快速计算

L ₂（ℝ）中的正交系统一旦以频谱方法实现，则如果其微分矩阵是倾斜对称且高度结构化的，则具有许多重要优势。这样的系统，其中微分矩阵是斜对称的，三对角的和不可约的，最近已得到充分表征。在本文中，我们进一步走了一步，强加了快速计算的额外要求：具体地说，可以在操作中以高精度计算扩展的前N个系数。我们认为两个设置，一个近似函数˚F直接在（-∞，∞）和其他近似[ ˚F（X）+ f（-x）] / 2和[ f（x）  -f（-x）] / 2分别在[0，∞）中。在每种情况下，我们证明存在一个由α，β  > − 1构成的正交系统的族，该族具有偏对称，三对角，不可约的微分矩阵，并且其系数可以作为修正函数的Jacobi多项式系数来计算。α，β  =±1/2的四种特殊情况由于系数可以使用快速正弦和余弦变换来计算，因此特别令人感兴趣。带状Toeplitz加上Hankel乘法运算符也可以用频谱方法表示可变系数。在傅立叶空间中，这些正交系统与Carlitz多项式的一个显然新的推广有关。©2020 Wiley Periodicals，Inc.