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The Hilbert transform of Schwartz distributions


Author: J. N. Pandey
Journal: Proc. Amer. Math. Soc. 89 (1983), 86-90
MSC: Primary 46F12; Secondary 44A15
MathSciNet review: 706516
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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

$\displaystyle \left\langle {Hf,\varphi } \right\rangle = \left\langle {f, - H\varphi } \right\rangle \forall \varphi \in H(\mathcal{D}).$

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.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1983-0706516-2
Keywords: Generalized integral transform, integral transform of distributions
Article copyright: © Copyright 1983 American Mathematical Society