approximating delta distribution

Question

I need some clarification for this. I know that we can approximate delta distribution by taking $f \in L^1$ by taking $f_j=j^n f(jx)$ when $\int f =c$ for some constant $c$. That`s o.k., but how can we do that when $\int f=0$ ? can anyone please explain this to me?

Thanks

Answer 1

When $f \in L^1$ then $k f(kx) \to F(+\infty)\delta(x)$ in the sense of distributions, where $F(x) = \int_{-\infty}^x f(y)dy$.

With $\varphi \in C^1$ and compactly supported then $$\lim_{k \to \infty}\int_{-\infty}^\infty k f(kx) \varphi(x) dx = -\lim_{k \to \infty}\int_{-\infty}^\infty F(kx) \varphi'(x) dx \\= -\int_0^\infty F(+\infty) \varphi'(x) dx = \varphi(0) F(+\infty)$$