What is...convolution?

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Goal.
I would like to tell you a bit about my favorite theorems, ideas or concepts in mathematics and why I like them so much.

This time.
What is...convolution? Or: Area, even without area.

Disclaimer.
Nobody is perfect, and I might have said something silly. If there is any doubt, then please check the references.

Slides.

TeX files for the presentation.

Thumbnail.

Main discussion.

Background material.

Mathematica.

Pictures used.

YouTube and co.

#analysis
#computeralgebra
#mathematics
  1. VisualMath
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While a bit more high-level, I've found the generalization of convolutions to groups (and the corresponding generalizations of Fourier transforms) to be very powerful. In my field, there is some notion of `noise', which is usually modelled as applying a certain group operation with a certain probability (where the group elements act on some set). Applying two such noise maps corresponds then really to the convolution of the probability distributions of the two maps. This is because the probability of applying an element g is equal to the sum over all probabilities corresponding to h and f such that h*f = g. In a lot of cases, it's a lot easier to understand (the composition of) noise after having done the Fourier transform, since for commutative groups you can think of your noise as ordinary multiplication.

Sqaarg