Inversion formula and range conditions for a linear system related with the multi‐interval finite Hilbert transform in L 2
Inversion formula and range conditions for a linear system related with the multi‐interval finite Hilbert transform in L 2
复制标题
L 2 中多区间有限希尔伯特变换相关线性系统的反演公式和范围条件
DOI：
10.1002/mana.201800567
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发表时间：
2021
影响因子：
1
通讯作者：
Tovbis, Alexander
中科院分区：
文献类型：
--
作者：
Katsevich, Alexander;Bertola, Marco;Tovbis, Alexander
给定不相交区间与函数、和矩阵一起，问题是求线性系统的L2解，其中是定义在上的有限希尔伯特变换的矩阵，是相应的特征函数的矩阵。由于我们可以解释为广义的多区间有限Hilbert变换，所以我们将解的公式称为“逆公式”，将解存在的充要条件称为“值域条件”。本文导出了两种特殊情况下的显式求逆公式和取值条件：a)矩阵Θ是对称正定的；b)Θ的所有项都等于1。我们还证明了解的唯一性，即我们的变换是内射的。在a)的情况下，即当矩阵Θ为正定时，用伴随矩阵黎曼-希尔伯特问题的解给出了求逆公式。在b)的情况下，我们将多区间问题归结为一个复本问题，然后用傅里叶变换来表示我们的解。我们还讨论了矩阵Θ的其他情况。
Givenndisjoint intervalsontogether withnfunctions,, and anmatrix, the problem is to find anL2solution,, to the linear system, where,is a matrix of finite Hilbert transforms withdefined on, andis a matrix of the corresponding characteristic functions on. Since we can interpret, 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 onncopies ofand then express our answers in terms of the Fourier transform. We also discuss other cases of the matrix Θ.