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Find a continuous function $f$ defined on $\mathbb{R}^d$ such that

(1) there is no polynomial $P$ in $d$ variables such that $|f(x)| \leq P(x)$ for all $x \in \mathbb{R}^d$.

(2) The distribution $\phi \mapsto \int \phi f dx$ is tempered.

I just learned tempered distribution, but I don't get the sense of how does a tempered distribution look like. So I don't have any idea of constructing such function.

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You will need something that oscillates wildly, yet defines a tempered distribution. The idea is to take something bounded (hence tempered), but with a very capricious derivative - and since derivatives of temepred distribution are tempered, we will obtain what we want.

Consider, for example, $f(x)=\sin(e^x)$. This is a bounded smooth function, hence $f$ is tempered: $f\in S'(\Bbb R)$.

Its derivative belongs to $S'$, too, as a derivative of tempered distribution.

Finally, $f' = e^x \cos(e^x) \in S'(\Bbb R)$, and it can not be bounded by a polynomial for obvious reasons - an exponent grows faster than any polynomial.

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  • $\begingroup$ Can you give some more insights for this example ? An infinitely oscillating term making $e^x$ tempered is interesting in deed. $\endgroup$
    – creative
    Sep 9, 2017 at 4:37

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