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Per Vognsen
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Late to the thread, but I wanted to quickly mention an identity that shows up for separable functions. Although this is a close cousin of your trivial identity and hardly theoretically deep, it turns out to be very useful in practice.

Let's take $\mathrm{R}^2$ as an example. If $f(x,y) = f_1(x)\ f_2(y)$ and $g(x,y) = g_1(x)\ g_2(y)$ then

$$f * g = (f_1\ f_2) * (g_1\ g_2) = (f_1 * g_1)\ (f_2 * g_2).$$

I abused notation a little to highlight the resemblance to distributivity.

This identity finds use in a folklore trick of image processing that is described here:

http://www.stereopsis.com/shadowrect/

Per Vognsen
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