1. Given a distribution $u \in \mathcal{D}'(\mathbb{R}^n)$ and a smooth function $f(x)$ on $\mathbb{R}^n$. Assume $f \cdot u \equiv 0$, then
   $$\operatorname{supp}(u) \subseteq Z(f) = \{x \in \mathbb{R}^n \mid f(x) = 0\}.$$

2. Consider the distribution $u = \sum_{n=0}^\infty c_n \delta_n$ on $\mathbb{R}$. Prove that $u \in \mathcal{S}'(\mathbb{R})$ if and only if there exist constants $C$ and $p$ such that for all non-negative integers $n$, we have
   $$|c_n| \leq C(1+n^p).$$

3. $u = \frac{1}{1+x^2}$;

4. $u = e^{-|x|}$;

5. $u = 1_{x>a}$, where $a \in \mathbb{R}$;

6. $u = \frac{1}{x+i0}$;

7. $u = P(x) 1_{x>0}$, where $P(x)$ is a polynomial with complex coefficients;

8. Assume $\alpha > -1$. Prove that
   $$\frac{1}{(i \xi + 0)^{1+\alpha}} \overset{\mathcal{D}'}{=} \lim_{\epsilon \to 0^+} (i \xi + \epsilon)^{-1-\alpha}$$
   defines a distribution on $\mathbb{R}$.

For $\alpha > -1$, use $\Gamma(z)$ (Gamma function) and
$$\frac{1}{(i \xi + 0)^{1+\alpha}}$$
to represent the Fourier transform of $u = \operatorname{pf} x_+^\alpha$;

9. $u = \operatorname{pf} x_+^\alpha$, where $\alpha < -1$ and $\alpha$ is not an integer;

10. $u = e^{-ix^2}$.