Let $T:\phi \mapsto \langle T_f,\phi\rangle =\int _{-\infty}^{\infty}f(x)\phi(x)dx$, where $\phi(x)$ is a test function, be a distribution. I would like to prove that the sequence of distributions: \begin{equation} \langle T_{f{_n}},\phi\rangle=\int _{-\infty}^{\infty}(tanh(nx)-1)\phi(x)dx \to \langle T_{f{}},\phi\rangle=-\int _{-\infty}^{0}\phi(x)dx+ \int _{0}^{\infty}\phi(x)dx \end{equation} with: \begin{equation} f_n(x)=tanh(nx) \end{equation} and \begin{equation} \lim_{x \to \pm \infty}f_n(x)=f(x)=\begin{cases} -1, & x<0 \\ 1, & x>0 \end{cases} \end{equation}
I know that I have to prove that the integral: \begin{equation} |\langle T_{f{n}},\phi\rangle-\langle T_{f{}},\phi\rangle|=\int_{0}^{\infty}(tanh(nx)-1)\phi(x)dx \to 0 \end{equation} In this case I must show somehow that:
- The sequence: $\{g_n\}_{n\in \mathbb{N}}=\{(tanh(nx)-1)\phi(x)\}_{n\in \mathbb{N}}$ converges uniformly to a function $f$, $\forall \phi \in D{\mathbb{(R)}}$.
- The functions $\{g_n\}_{n\in \mathbb{N}}$ are Riemann integrable.
Are the above right? If yes, how am I supposed to prove the uniform convergence? I do know that they are Riemann integrable since they are continuous and supp$g_n=[a,b]$ a compact interval. Can you please guide me throught the procedure since it is the first kind of exercise that I am trying to do on distributions and I would like to have an example to start with. Thank you all!
You have that $f_n \phi \to f \phi$ pointwise
As $|f_n (x) \phi(x)| \leq |\phi(x)|$, $f_n \phi$ is dominated by $|\phi|$, that is an integrable fontion.
Hence, all the hypothesis of the Lebesgue's Dominated Convergence Theorem are satisfied, and this imply that
$$\lim_{n\to +\infty} \int_{\mathbb{R}} |f_n (x) \phi(x)-f(x)\phi(x)| dx = 0$$