Which of the functions below define a tempered distribution? The functions in a, b, c are defined in $\mathbb{R}$.
a) $f(x)=x^3+3x$ $$|T(\phi)| \leq \int_{\mathbb{R}} |(x^3+3x)\phi(x)|dx=\int_{\mathbb{R}} |x(x^2+3)\phi(x)|dx \leq \|x(x^2+3)\phi\|_\infty \times C$$
So f(x) is a tempered distribution where $C=1$.
b) $S(x)=\sum_{n=-\infty}^\infty (-1)^n \chi_{(n,n+1)}(x)$ $$|T(\phi)| \leq \int_{\mathbb{R}} \sum_{n=-\infty}^\infty |(-1)^n \chi_{(n,n+1)}(x)\phi(x)|dx=\sum_{n=-\infty}^\infty \int_n^{n+1} |\phi(x)|dx \leq \|\phi\|_\infty \sum_{n=-\infty}^\infty 1=\infty$$
So $S(x)$ isn't a tempered distribution.
c) $g(x)=(x-1)^\frac{-1}{3}$ $$|T(\phi)| \leq \int_{\mathbb{R}} |\frac{1}{(x-1)^\frac{1}{3}}\phi(x)|dx \leq \|\phi\|_\infty \times 0=0$$
So $g(x)$ is a tempered distribution where $C=0$.
d) $h(x)=e^{|x|^2-|x|}$ $$|T(\phi)| \leq \int_{{\mathbb{R}}^n} |e^{|x|^2-|x|}\phi(x)|\,dx \leq \|\phi\|_\infty \int_{{\mathbb{R}}^n} |e^{|x|^2-|x|}|\,dx$$
How do I continue from here?
e) $k(x)=x_1 x_2 e^{-|x|}$
Help
f) $\mu(x)=|x|\frac{\sin(x_1)}{x_1}$
Help