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Some properties of a modified Hilbert transform

Matteo Ferrari

TL;DR

The paper analyzes the modified Hilbert transform $\mathcal{H}_T$ on $(0,T)$ and proves that it is exactly the canonical Hilbert transform applied to a suitably defined odd periodic extension: $\mathcal{H}_T \varphi = -\mathcal{H}\widetilde{\varphi}$ in $L^2(0,T)$. This core identity is established via three independent proofs—through the integral representation, the periodic Hilbert transform framework, and Fourier-series methods—each confirming the same relation. Consequences include a direct inversion formula $\mathcal{H}(\mathcal{H}_T\varphi)=\varphi$, explicit kernel representations for $\mathcal{H}_T$ and $\mathcal{H}$-based kernels, and an integral representation of $\mathcal{H}_T\varphi$ in terms of $\partial_t\widetilde{\varphi}$ with boundary terms. By linking $\mathcal{H}_T$ to the well-studied $\mathcal{H}$, the results enhance the theoretical understanding of the modified operator and its role in space-time discretizations for PDEs.

Abstract

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 $\mathcal{H}$.

Some properties of a modified Hilbert transform

TL;DR

The paper analyzes the modified Hilbert transform on and proves that it is exactly the canonical Hilbert transform applied to a suitably defined odd periodic extension: in . This core identity is established via three independent proofs—through the integral representation, the periodic Hilbert transform framework, and Fourier-series methods—each confirming the same relation. Consequences include a direct inversion formula , explicit kernel representations for and -based kernels, and an integral representation of in terms of with boundary terms. By linking to the well-studied , the results enhance the theoretical understanding of the modified operator and its role in space-time discretizations for PDEs.

Abstract

Recently, Steinbach et al. introduced a novel operator , 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 and the canonical Hilbert transform . Specifically, we prove the relationship , where and is a suitable extension of over the entire . By leveraging this crucial result, we derive some properties of , including a new inversion formula, that emerge as immediate consequences of well-established findings on .
Paper Structure (10 sections, 3 theorems, 33 equations, 2 figures)

This paper contains 10 sections, 3 theorems, 33 equations, 2 figures.

Table of Contents

  1. Introduction and main result
  2. Proofs of the main result
  3. Proof based on the integral representation
  4. Proof based on the Hilbert transform for periodic functions
  5. Proof based on Fourier series
  6. Consequences of the main result
  7. Inversion formula
  8. Alternative formula
  9. Integral representation
  10. Acknowledgements

Key Result

Lemma 1

For $\varphi \in L^2(0,T)$, the operator $\mathcal{H}_T$ allows the integral representation as Cauchy principal value integral. Moreover, if $v \in H^1(0,T)$ it holds for $t \in (0,T)$ as a weakly singular integral.

Figures (2)

  • Figure 1: For $\varphi(x) = x^2$, $\varphi : [0,T] \to \mathbb{R}$, it is plotted $\bar{\varphi} : \mathbb{R} \to \mathbb{R}$ defined in \ref{['barphi']}.
  • Figure 2: For $\varphi(x) = x^2$, $\varphi : [0,T] \to \mathbb{R}$, it is plotted $\widetilde{\varphi} : \mathbb{R} \to \mathbb{R}$ defined in \ref{['tphi']}.

Theorems & Definitions (5)

  • Lemma 1
  • Theorem 1
  • Theorem 2
  • Example 1
  • Example 2