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I have a question about the second answer here. The last step should be

$$ \int_{-\infty}^{\infty} F(w) \delta (w-\hat w)\mathrm{d}w = \frac{1}{2\pi}\int_{-\infty}^\infty F(w) \left(\int_{-\infty}^\infty e^{i(w-\hat{w})t}\,\mathrm{d}t \right) \,\mathrm{d}w$$

After derivation of the equation, it is:

$$\frac{1}{2\pi}\int_{-\infty}^\infty e^{i(w-\hat{w})t}\,\mathrm{d}t= \delta (w- \hat w) $$

Is that correct?

What if there were sums instead of integrals, please? How to get

$$\frac{1}{2\pi}\sum_{t} e^{i(w-\hat{w})t}= \delta (w- \hat w) $$ ?

J.G.
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Anna-Kat
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  • Relevant. – J.G. Sep 27 '22 at 19:25
  • Which of the equations in the link do you excatly mean, please? – Anna-Kat Sep 27 '22 at 19:26
  • To be honest I think you should read the article as a whole, as well as the Fejér kernel article to which it links, to familiarize yourself with not only the equations but relevant concepts. The important point is that we acquire a family of functions integrating to $1$ that pointwise approximate $\delta(x)$. – J.G. Sep 27 '22 at 19:33
  • I went through the article, but I still do not get it. – Anna-Kat Sep 28 '22 at 04:02

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