i'm not sure if this should be obvious, but it surprised me to find that both are equal to πpic.twitter.com/973xR17ilx
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I think this simply doesn't apply here. The sinc function oscillates with the same frequency on its whole domain. In no way does its discrete sum approximate its integral, except for the fact that they happen to be equal.
The approximation gets better the further you go out, though. I *think* this is because the peaks and troughs then tend to just their heights cause ω-> infinity
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I mean, any explanation of this needs to explain how the errors for small x get exactly compensated. Arguing about the large x behavior here isn't going to tell you about why the sum and integral exactly match.pic.twitter.com/OGTE7eTxeW
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It is *very* interesting to note that when you actually properly take the limit as the rectangles get smaller, the result is insensitive to the size of the rectangles (so long as they don't get too big).
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