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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This feels strictly... not true? The sum of a function over a domain only approaches its Riemann sum when the rectangle widths vanish relative to the characteristic scale on which the function varies. This simply doesn't happen here. Unless I'm missing something..?
i was struggling with that too, but thought maybe i was mistaken.. What's your reasoning?
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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.
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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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