Some properties of a modified Hilbert transform

Recently, Steinbach et al. introduced a novel operator HT: L²(0,T) → L²(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 HT and the canonical Hilbert transform H. Specifically, we prove the relationship HT φ = -H φ̃, where φ ∈ L²(0,T) and φ̃ is a suitable extension of φ over the entire R. By leveraging this crucial result, we derive some properties of HT, including a new inversion formula, that emerge as immediate consequences of well-established findings on H.