Problem on temperate distribution theory

Question

Problem: Give an example of an infinitely differentiable function f on $\mathbf{R}$ such that (1): there does not exist a polynomial P on $\mathbf{R}$ such that $|f(x)| \le |P(x)|, x \in \mathbf{R}$ and (2): the mapping $\mathbf{S}(\mathbf{R}) \ni \varphi \rightarrow \int f(x) \varphi(x)dx$ defines a temperate distribution.

Denote by $\mathbf{S}$ the Schwartz space of infinitely differentiable functions $\phi(x)$ such that $|\frac{\partial^k \varphi(x)}{\partial x^k}| \le C_{m,k}(1+|x|)^{-m}$ for any $k$ and any positive integer $m$. Introduce the norms $|[\varphi]|_{m,S}=\textbf{sup}_{x \in \mathcal{R}^n} (1+|x|)^m \sum_{|k|=0}^m |\frac{\partial^k \varphi(x)}{\partial x^k}|$. We say that $\varphi_n \rightarrow \varphi in S$ if $|[\varphi_n-\varphi]|_{m,S} \rightarrow 0$ as $n \rightarrow \infty$ for all $m=1,2,...,$

Definition: Linear continuous functional on S is called a tempered distribution. The linear space of tempered distributions is denoted by $S'$.

Theorem 9.1: For any tmepered distrubition f, there exist an integer m and a constant C such that $|f(\varphi)| \le C |[\varphi]|_{m,S} \forall \varphi \in S$.

Theorem 9.2: Any $f \in S'$ can be represented in the following form: $f=\sum_{|k|=0}^{m_1} \frac{\partial^k \varphi(x)}{\partial x^k} f_k$, where $f_k$ are regular functions in $S'$ corresponding to continuous functions $f_k(x)$ satisfying the estimates $|f_k(x)| \le C_k(1+|x|)^{m_2}$. Here $m_1,m_2$ are integers.

Question: I have no clue how to approach this problem. Any help is appreciated.

Answer 1

Pick your favorite bump function $\Psi(x)$ then define $$ \Psi_{p,w,h}(x)=h\Psi\left(\frac{x-p}{w}\right)\ . $$ Here $p$ allows to pick the position of where you want it translated. $h$ allows you to adjust the height and $w$ controls the width. You can construct $f$ in the form $$ f(x)=\sum_{n=0}^{\infty}\Psi_{n,w_n,h_n}(x) $$ for suitable sequences $(w_n)$ and $(h_n)$. First pick the $h_n$ so they grow faster than any polynomial, e.g., $h_n=e^n$. Then pick the widths small enough so that 1) the bumps do not touch and the areas of the bumps are so small so the total area under $f$ is finite. Then $f(x)dx$ is a bounded measure and thus a temperate measure and it will define a temperate distribution.

Answer 2

Let $F(x)=\sin e^x$. This is clearly both continuous and bounded, so it defines a tempered distribution. And if $F$ defines a tempered distribution, then so does its derivative! Thus, $f(x)=F'(x)=e^x \cos e^x$ defines a tempered distribution. But $|f(x)|$ grows exponentially and is therefore not bounded by any polynomial.