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 ϕe, where ϕ ∈ L2(0, T ) and ϕe 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 .
Some properties of a modified Hilbert transform
Matteo Ferrari
2024-01-01
Abstract
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 ϕe, where ϕ ∈ L2(0, T ) and ϕe 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 .I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
