Recently, Steinbach et al. introduced a novel operator $\mathcal{H}_T: L^2(0,T) \to L^2(0,T)$, known as the modified Hilbert transform. This operator has shown its significance in space-time formulations related to the heat and wave equations. In this paper, we establish a direct connection between the modified Hilbert transform $\mathcal{H}_T$ and the canonical Hilbert transform $\mathcal{H}$. Specifically, we prove the relationship $\mathcal{H}_T \varphi = -\mathcal{H} \tilde{\varphi}$, where $\varphi \in L^2(0,T)$ and $\tilde{\varphi}$ is a suitable extension of $\varphi$ over the entire $\mathbb{R}$. By leveraging this crucial result, we derive some properties of $\mathcal{H}_T$, including a new inversion formula, that emerge as immediate consequences of well-established findings on $\mathcal{H}$.

翻译：近期，Steinbach等人引入了一个新的算子$\mathcal{H}_T: L^2(0,T) \to L^2(0,T)$，称为修正希尔伯特变换。该算子在热传导方程和波动方程的时空领域求解中展现了重要价值。本文建立了修正希尔伯特变换$\mathcal{H}_T$与经典希尔伯特变换$\mathcal{H}$之间的直接联系。具体而言，我们证明了关系式$\mathcal{H}_T \varphi = -\mathcal{H} \tilde{\varphi}$，其中$\varphi \in L^2(0,T)$，而$\tilde{\varphi}$是$\varphi$在全空间$\mathbb{R}$上的适当延拓。借助这一关键结论，我们推导出$\mathcal{H}_T$的一些性质，包括一个新的反演公式，这些性质均为经典希尔伯特变换相关结论的直接推论。