Given n disjoint intervals $I_j$ on $\mathbb{R}$ together with n functions ${\psi }_j\in L^2\left(I_j\right)$, $j=1,\cdots n$, and an $n\times n$ matrix $\mathrm{\Theta }=\left({\theta }_{jk}\right)$, the problem is to find an L² solution $\vec{\phi }=\mathrm{Col}({\phi }_1,\cdots ,{\phi }_n)$, ${\phi }_j\in L^2\left(I_j\right)$, to the linear system $\chi \mathrm{\Theta }\mathcal{H}\vec{\phi }=\vec{\psi }$, where $\vec{\psi }=\mathrm{Col}({\psi }_1,\cdots ,{\psi }_n)$, $\mathcal{H}=\mathrm{diag}({\mathcal{H}}_1,\cdots ,{\mathcal{H}}_n)$ is a matrix of finite Hilbert transforms with ${\mathcal{H}}_j$ defined on $L^2\left(I_j\right)$, and $\chi =\text{diag}({\chi }_1,\cdots ,{\chi }_n)$ is a matrix of the corresponding characteristic functions on $I_j$. Since we can interpret $\chi \mathrm{\Theta }\mathcal{H}\vec{\phi }$, as a generalized multi-interval finite Hilbert transform, we call the formula for the solution as “the inversion formula” and the necessary and sufficient conditions for the existence of a solution as the “range conditions”. In this paper we derive the explicit inversion formula and the range conditions in two specific cases: a) the matrix Θ is symmetric and positive definite, and; b) all the entries of Θ are equal to one. We also prove the uniqueness of solution, that is, that our transform is injective. In the case a), that is, when the matrix Θ is positive definite, the inversion formula is given in terms of the solution of the associated matrix Riemann–Hilbert Problem. In the case b) we reduce the multi interval problem to a problem on n copies of $\mathbb{R}$ and then express our answers in terms of the Fourier transform. We also discuss other cases of the matrix Θ.

中文翻译：

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L2中多区间有限希尔伯特变换相关的线性系统的反演公式和范围条件

给定n 个不相交的区间${\mathrm{一世}}_j$ 在 $\mathbb{电阻}$连同n个函数${\psi }_j\in {升}^2\left({\mathrm{一世}}_j\right)$, $j=1,\cdots n$， 和 $n\times n$ 矩阵 $\mathrm{\Theta }=\left({\theta }_{j克}\right)$, 问题是找到一个L ²解$\vec{\phi }=\mathrm{上校}({\phi }_1,\cdots ,{\phi }_n)$, ${\phi }_j\in {升}^2\left({\mathrm{一世}}_j\right)$, 线性系统 $\chi \mathrm{\Theta }\mathcal{H}\vec{\phi }=\vec{\psi }$， 在哪里 $\vec{\psi }=\mathrm{上校}({\psi }_1,\cdots ,{\psi }_n)$, $\mathcal{H}=\mathrm{诊断}({\mathcal{H}}_1,\cdots ,{\mathcal{H}}_n)$ 是有限希尔伯特变换的矩阵，其中 ${\mathcal{H}}_j$ 定义于 ${升}^2\left({\mathrm{一世}}_j\right)$， 和 $\chi =\text{诊断}({\chi }_1,\cdots ,{\chi }_n)$ 是对应特征函数的矩阵 ${\mathrm{一世}}_j$. 既然我们可以解释$\chi \mathrm{\Theta }\mathcal{H}\vec{\phi }$，作为广义的多区间有限希尔伯特变换，我们将解的公式称为“反演公式”，将解存在的充要条件称为“范围条件”。在本文中，我们推导出了两种特定情况下的显式反演公式和范围条件：a) 矩阵 Θ 是对称正定矩阵，并且；b) Θ 的所有条目都等于 1。我们还证明了解的唯一性，即我们的变换是单射的。在 a) 情况下，即当矩阵 Θ 为正定矩阵时，根据关联矩阵 Riemann-Hilbert 问题的解给出反演公式。在情况 b) 中，我们将多区间问题简化为n 个副本上的问题$\mathbb{电阻}$然后用傅立叶变换表达我们的答案。我们还讨论了矩阵 Θ 的其他情况。