Some properties of a modified Hilbert transform

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 ${\mathcal{H}}_T\phi =-\mathcal{H}\tilde{\phi }$, where $\phi \in L^2(0,T)$ and $\tilde{\phi }$ is a suitable extension of $\phi$ 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 $\mathcal{H}$.