# Improper integral of a sinc function

I am trying to show that the sinc function $$\frac{\sin(ax)}{x}$$ behaves like a delta distribution when $$\lim({a \to \infty})$$. I can show that $$\int_{-\infty}^{\infty} \frac{\sin(x)}{x}=\pi$$ Therefore, $$\lim_{a \to \infty}\int_{-\infty}^{\infty}f(x)\frac{\sin(ax)}{x}dx = \lim_{a \to \infty}\int_{-\infty}^{\infty}f(t/a)\frac{\sin(t)}{t}dt = \int_{-\infty}^{\infty}f(0)\frac{\sin(t)}{t}dt = \pi f(0)$$ where I substituted $$x=t/a$$. However, I am unable to show that $$\lim_{a \to \infty}\int_{0^+}^{\infty}f(x)\frac{\sin(ax)}{x}dx = 0$$

P.S. If someone can refer me to a text which treats this issue, it will be appreciated. (Unimportant details: I stumbled across this issue while exploring the Fourier transform for a constant function. I know that inverse Fourier can be used to get around the problem but I would like a rigorous treatment of sinc function as a delta distribution)

• Have you seen math.stackexchange.com/questions/744943/…? – Martín-Blas Pérez Pinilla Apr 3 at 13:00
• Yes, I have. That is a very different question. My question is asking how to solve the integral with 0 excluded from the domain. This is a different property. Proving all 3 properties will show that this distribution yields the exact same result as the delta function. – SSB Apr 3 at 13:08