A note on a modified Hilbert transform

Abstract

The Hilbert transform $\mathcal{H}$ is a useful tool in the mathematical analysis of time-dependent partial differential equations in order to prove coercivity estimates in anisotropic Sobolev spaces in case of a bounded spatial domain Ω, but an infinite time interval $(0,\mathrm{\infty })$. Instead, a modified Hilbert transform ${\mathcal{H}}_T$ can be used if we consider a finite time interval $(0,T)$. In this note we prove that the classical and the modified Hilbert transformations differ by a compact perturbation, when a suitable extension of a function defined on a bounded time interval $(0,T)$ onto $\mathbb{R}$ is used. This result is important when we deal with space–time variational formulations of time-dependent partial differential equations, and for the implementation of related space–time finite and boundary element methods for the numerical solution of parabolic and hyperbolic equations with the heat and wave equations as model problems, respectively.

Keywords:

• Hilbert transform
• modified Hilbert transform
• compact perturbation

2020 Mathematics Subject Classification:

• 44A15

Disclosure statement

No potential conflict of interest was reported by the author(s).