In this paper, we explore orthogonal systems in \(\mathrm {L}_2({\mathbb R})\) which give rise to a real skew-symmetric, tridiagonal, irreducible differentiation matrix. Such systems are important since they are stable by design and, if necessary, preserve Euclidean energy for a variety of time-dependent partial differential equations. We prove that there is a one-to-one correspondence between such an orthonormal system \(\{\varphi _n\}_{n\in {\mathbb Z}_+}\) and a sequence of polynomials \(\{p_n\}_{n\in {\mathbb Z}_+}\) orthonormal with respect to a symmetric probability measure \(\mathrm{d}\mu (\xi ) = w(\xi ){\mathrm {d}}\xi \). If \(\mathrm{d}\mu \) is supported by the real line, this system is dense in \(\mathrm {L}_2({\mathbb R})\); otherwise, it is dense in a Paley–Wiener space of band-limited functions. The path leading from \(\mathrm{d}\mu \) to \(\{\varphi _n\}_{n\in {\mathbb Z}_+}\) is constructive, and we provide detailed algorithms to this end. We also prove that the only such orthogonal system consisting of a polynomial sequence multiplied by a weight function is the Hermite functions. The paper is accompanied by a number of examples illustrating our argument.

中文翻译：

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具有斜对称微分矩阵的正交系统

在本文中，我们探索\（\ mathrm {L} _2（{\ mathbb R}）\）中的正交系统，该系统产生了一个真正的斜对称，对角线，不可约的微分矩阵。这样的系统很重要，因为它们在设计上很稳定，并且在必要时可以为各种随时间变化的偏微分方程保留欧几里得能量。我们证明了这样的正交系统\（\ {\ varphi _n \} _ {n \ {\ mathbb Z} _ +} \中）与多项式序列\（\ { p_n \} _ {n \ in {\ mathbb Z} _ +} \）关于对称概率测度\（\ mathrm {d} \ mu（\ xi）= w（\ xi）{\ mathrm {d }} \ xi \）。如果实线支持\（\ mathrm {d} \ mu \），则该系统在\（\ mathrm {L} _2（{\ mathbb R}）\） ; 否则，它在带限函数的Paley-Wiener空间中是密集的。从\（\ mathrm {d} \ mu \）到\（\ {\ varphi _n \} _ {n \ in {\ mathbb Z} _ +} \）的路径是建设性的，为此我们提供了详细的算法结束。我们还证明，由多项式序列乘以权重函数组成的唯一此类正交系统是Hermite函数。本文附带了许多说明我们的论点的例子。