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In this video, it is stated that Parseval's Identity "is how we go from discrete to continuous." However, I have not been able to find any material that expands on this use of Parseval's Identity. I have discovered that it is the inner product version of the Pythagorean Theorem.

What does this quote that Parseval's Identity "is how we go from discrete to continuous" mean? Is Parseval's Identity a bridge between recurrence relations and differential equations somehow? In this vein, is it related to time-scale calculus or this application of generating functions?

user10478
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  • I suspect it's because parseval's identity relates a "continuous" object (namely the integral $\int |f|^2$) to an associated "discrete" object (namely the sum of the squares of the fourier coefficients $\sum |\hat{f}|^2$). Frequently this theorem is used when we have information about $\hat{f}$ (a discrete object, since it's a function on $\mathbb{Z}$) and we want to convert that into information about $f$ (a continuous object). – Chris Grossack Feb 28 '22 at 21:04

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