Inversion formula and range conditions for a vector multi-interval finite Hilbert transform in $L^2$

TL;DR

This paper derives explicit inversion formulas and range conditions for a vector multi-interval finite Hilbert transform with specific matrix structures, proving solution uniqueness and connecting to Riemann-Hilbert problems.

Contribution

It provides the first explicit inversion formulas and range conditions for the vector finite Hilbert transform in cases of symmetric positive definite and uniform matrices, including solution uniqueness.

Findings

Explicit inversion formulas derived for specific matrix cases.

Range conditions established for solution existence.

Proved injectivity of the transform.

Abstract

Given $n$ disjoint intervals $I_j$, on $\mathbb R$ together with $n$ functions $\psi_j\in L^2(I_j)$, $j=1,\dots n$, and an $n\times n$ matrix $\Theta$, the problem is to find an $L^2$ solution $\vec \varphi= {\rm Col} (\varphi_1,\dots, \varphi_n)$, $\varphi_j \in L^2(I_j)$ to the linear system $\chi\Theta \mathcal H \vec \varphi = \vec\psi$, where $\mathcal H = {\rm diag} (\mathcal H_1 ,\dots, \mathcal H_n)$ is a matrix of finite Hilbert transforms and $\chi=\text{diag}(\chi_1,\dots,\chi_n)$ is a matrix of the corresponding characteristic functions on $I_j$, and $\vec \psi={\rm Col} (\psi_1,\dots,\psi_n)$. Since we can interpret $\chi\Theta \mathcal H \vec \varphi$ as a generalized vector 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 $\Theta$ is symmetric and positive definite, and; b) all the entries of $\Theta$ are equal to one. We also prove the uniqueness of solution, that is, that our transform is injective. When the matrix $\Theta$ is positive definite, the inversion formula is given in terms of the solution of the associated matrix Riemann-Hilbert Problem. We also discuss other cases of the matrix $\Theta$.