Discrete convolutions, from probability to image processing and FFTs.
Video on the continuous case:    • Convolutions | Why X+Y in probability...  
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Other videos I referenced

Live lecture on image convolutions for the MIT Julia lab
   • Convolutions in Image Processing | We...  

Lecture on Discrete Fourier Transforms
   • What is a Discrete Fourier Transform?...  

Reducible video on FFTs
   • The Fast Fourier Transform (FFT): Mos...  

Veritasium video on FFTs
   • The Remarkable Story Behind The Most ...  

A small correction for the integer multiplication algorithm mentioned at the end. A “straightforward” application of FFT results in a runtime of O(N * log(n) log(log(n)) ). That log(log(n)) term is tiny, but it is only recently in 2019, Harvey and van der Hoeven found an algorithm that removed that log(log(n)) term.

Another small correction at 17:00. I describe O(N^2) as meaning "the number of operations needed scales with N^2". However, this is technically what Theta(N^2) would mean. O(N^2) would mean that the number of operations needed is at most constant times N^2, in particular, it includes algorithms whose runtimes don't actually have any N^2 term, but which are bounded by it. The distinction doesn't matter in this case, since there is an explicit N^2 term.

Thanks to these viewers for their contributions to translations
Hebrew: Omer Tuchfeld
Italian: Emanuele Vezzoli
Vietnamese: lkhphuc

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Music by Vincent Rubinetti.
https://www.vincentrubinetti.com/

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Timestamps
0:00 - Where do convolutions show up?
2:07 - Add two random variables
6:28 - A simple example
7:25 - Moving averages
8:32 - Image processing
13:42 - Measuring runtime
14:40 - Polynomial multiplication
18:10 - Speeding up with FFTs
21:22 - Concluding thoughts

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