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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)

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  • $\begingroup$ Have you seen math.stackexchange.com/questions/744943/…? $\endgroup$ – Martín-Blas Pérez Pinilla Apr 3 at 13:00
  • $\begingroup$ 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. $\endgroup$ – SSB Apr 3 at 13:08

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