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I am reading Rudin's Functional Analysis and got quite confused by his proof of Thm 8.5, that is, the existence of fundamental solutions for differential operator $P(D)$, where $P$ is a polynomial.

The proof begins with defining a norm on the space of test functions $D(\mathbb{R}^d)$ by \begin{equation}\|\psi\|=\int_{T^d}\int_{\mathbb{R}^d}|\hat{\psi}(t+r\omega)|dm_d(t)d\sigma_d(\omega),\end{equation}where $r>0$ is fixed, $T^d$ is the torus in $\mathbb{C}^d$, $\sigma_d$ the Haar measure for this torus, and $m_d$ is the Lebesgue measure for $\mathbb{R}^d$.

Then the proof goes on well, and this norm is used to prove the existence of a distribution, that is, our fundamental solution.

However, I do not know how Rudin thinks about this strange norm. It reminds of the norm on Sobolev spaces but you do not have the integration over torus there.

Can somebody give some reference or background information about this norm?

Thanks!

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  • $\begingroup$ I suspect the norm was cooked up for this proof. Have you checked in the Notes and Comments at the back of the book? $\endgroup$ – Harald Hanche-Olsen Oct 19 '12 at 10:08

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