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I'm trying to find nonzero functions $f:\Bbb R^2\to\Bbb R$ such that the following holds:

$$f(x,z)^2=\int_\Bbb R f(x,y)f(y,z)dy$$

but couldn't find any. Any help on methods of finding solutions, or even finding one solution would be of great help. Thanks.

Garmekain
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1 Answers1

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$$f(x,y) = g(x)g(y)\delta(x-y)$$ almost works

If you look at the discrete version with matrices, you get the equation

$$ A \otimes A = A^2 $$ where $\otimes$ denotes entrywise product. The above is satisfied by diagonal matrices.

However in the continuous case with $f(x,y) = g(x)g(y)\delta(x-y)$ you would get a squared Dirac delta function on the LHS and a simple Dirac delta function on the RHS, so the match isn't perfect...

phaedo
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