Growth conditions for membership in S'

Prove that $\delta_{0} \in \mathcal{S}^{\prime }(\mathbb {R})$ and $\delta_{0}^{(k)} \in \mathcal{S}^{\prime }(\mathbb {R})$ for all $k \geq 1$.

Prove that $e^{x^{2}} \in \mathcal{D}^{\prime }(\mathbb {R})$, but $e^{x^{2}}$ does not generate a regular tempered distribution from the space $S^{\prime }(\mathbb {R})$.

Let $f \in \mathcal{S}^{\prime }(\mathbb {R})$ and $P$ be a polynomial. Define the functional $\left\langle P f, \varphi \right\rangle = \left\langle f, P \varphi \right\rangle , \varphi \in \mathcal{S}(\mathbb {R})$. Prove that $P f \in \mathcal{S}^{\prime }(\mathbb {R})$.

Prove that if a function $f$ has the form $f(x) = g(x)\left(1+\left|x\right|^{n}\right)$, where $g \in L_{1}(\mathbb {R})$ and $n \in \mathbb {N} \cup \left\{ 0\right\}$, then $f$ generates a generalized function $F_{f} \in S^{\prime }$.

Prove the stated assertion. The generalized function generated by the locally integrable function $e^{x}$ cannot be extended from the set of test functions $D$ to the set of test functions $S$, i.e., it cannot be extended to an element of the space $S^{\prime }$.

Prove the stated assertion. If a function $f(x)$ is absolutely integrable over the whole real line, then the functional $f$ defined by the formula

belongs to the space $S^{\prime }$.

Prove the stated assertion. If a function $f: R \rightarrow C$ is locally integrable and satisfies the estimate $\left|f(x)\right| \leqslant c\left|x\right|^{k}$, where $x$ and $k$ are nonnegative constants, then the functional $f$ defined by the formula

belongs to the space $S^{\prime }$.

Prove the stated assertion. If $f \in S^{\prime }$, and the function $\psi$ satisfies the conditions of problem 109, then $\psi f \in S^{\prime }$.

Suppose $f(x) = e^{x}, g(x) = e^{x} \cos \left(e^{x}\right)$, on the real line. Show that $g$ is a tempered distribution but that $f$ is not.