The Hilbert transform of Schwartz distributions

Abstract:

Let $\mathcal {D}$ be the Schwartz space of infinitely differentiable complex-valued functions defined on the real line with compact supports equipped with the usual topology. Assume $H(\mathcal {D})$ to be the space of ${C^\infty }$ functions defined on the real line whose every element is the Hilbert transform of an element of $\mathcal {D}$. We equip the space $H(\mathcal {D})$ with an appropriate topology and show that the classical Hilbert transformation $H$, defined by $Hf = P\int _{ - \infty }^\infty {f(t)/(t - x)dt}$, is a homeomorphism from $\mathcal {D}$ onto $H(\mathcal {D})$. The Hilbert transform $Hf$ of $f \in \mathcal {D}’$ is then defined to be an element of $H’(\mathcal {D})$ given by the relation \[ \left \langle {Hf,\varphi } \right \rangle = \left \langle {f, - H\varphi } \right \rangle \forall \varphi \in H(\mathcal {D}).\] It then follows that -$- {H^2}f/{\pi ^2} = f\forall f \in \mathcal {D}’$. Applications of our results in solving some singular integral equations are also discussed.