Prove that Sequence $\phi _n(x)=n\phi(nx)$ is delta sequence

Question

Let $\phi : \Bbb R \to [0, \infty)$ be such that $\int_{-\infty}^{\infty} \phi (x) dx=1 $. Define $\phi _n(x)=n\phi(nx)$. Show that $\{\phi _n(x)\}_{n\in \Bbb N}$ is a delta sequence.

Attempt: To show this I need to show

$$\lim_{n\to\infty}(\phi_n,\psi)=(\delta,\psi)=\psi(0), \quad \forall \psi\in \mathcal{D}(\mathbb{R}).$$

$(\phi_n,\psi)=\int_{-\infty}^{\infty}n\phi(nx)\psi(x)dx.$

After Substitution $y=nx$, $\int_{-\infty}^{\infty}n\phi(nx)\psi(x)dx=\int_{-\infty}^{\infty}\phi(y)\psi\left(\frac{y}{{n}}\right)dy.$ After that I dont know, what have to do.

Answer 1

Hint

For all $y$ $\phi(y) \psi \left( \frac{y}{n}\right) \rightarrow \phi(y) \psi(0)$ and use dominated convergence theorem.

Answer 2

Hint: let $f_n\colon y\mapsto \phi\left(y\right)\psi\left(y/n\right)$.

• What is the pointwise limit of $\left(f_n\right)_{n\geqslant 1}$?
• Can you bound $\left\lvert f_n(y)\right\rvert$ by a quantity independent of $n$ and integrable in $y$?