Limit in a sense of distributions

Question

How to find $\lim_{a\rightarrow\infty} f_a$ in $D'(R)$, for $a>0$, where $f_a:R\rightarrow R$ is defined by

$f_a(x)=\begin{cases}\frac{\sin{ax}}{x}&x\neq 0 \\0&x=0\end{cases}$

Thanks in advance.

Answer 1

You want to calculate for $\phi\in D(R)$ $$ \lim_{a\to\infty} \int_R \frac{\sin ax}{x} \phi(x) dx $$ which is well-defined as the integrand is continuous. Substituting $y=ax$, the integral becomes $$ \int_R \frac{\sin y}{y}\phi(\frac{y}{a}) dy. $$ As $\phi(\frac{y}{a}) = \phi(0) + \frac{y}{a}\phi'(\xi_y)$ for some $\xi_y$, you have $$ \int_R \frac{\sin y}{y}\phi(\frac{y}{a}) dy = \pi\phi(0) + \frac{1}{a} \int_R \sin y \phi'(\xi_y) dy.$$ The latter integral is bounded independently of $a$, so you conclude $$ \lim_{a\to\infty} \frac{\sin ax}{x} = \pi\delta.$$