Some properties of a modified Hilbert transform
<|reference_start|>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 $\mathcal{H}_T \varphi = -\mathcal{H} \tilde{\varphi}$, 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 $\mathcal{H}$.<|reference_end|>
@article{ferrari2024some,
    title={Some properties of a modified Hilbert transform},
    author={Matteo Ferrari},
    journal={arXiv preprint arXiv:2404.02609},
    year={2024},
    archivePrefix={arXiv},
    eprint={2404.02609},
    primaryClass={math.CA cs.NA math.NA}
}