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Explicit isomorphism between $L^2(\mathbb{R}^2)$ and $L^2(\mathbb{R})$?
As Hilbert spaces, $L^2(\mathbb{R}^2)$ and $L^2(\mathbb{R})$ are isomorphic. Of course the isomoprhism is vastly not unique. I wonder if there are any particularly nice explicit isomorphisms. E.g. I ...
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18
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Complex structure on $L^2(\mathbb R)$ generalizing the Hilbert transform
The Hilbert transform on the real Hilbert space $L^2(\mathbb R)$ is the singular integral operator
$$
\mathcal H(f)(x) := \frac{1}{\pi} \int_{-\infty}^\infty \frac{1}{x-y} f(y) dy.
$$
It satisfies $\...
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