The Hilbert transform of Schwartz distributions
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- by J. N. Pandey PDF
- Proc. Amer. Math. Soc. 89 (1983), 86-90 Request permission
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.References
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Additional Information
- © Copyright 1983 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 89 (1983), 86-90
- MSC: Primary 46F12; Secondary 44A15
- DOI: https://doi.org/10.1090/S0002-9939-1983-0706516-2
- MathSciNet review: 706516