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$\forall y \in \mathbb{R}, \int_{-\infty}^{\infty} f(x)f(x-y)dx=f(y)$

I also know that $\int_{-\infty}^\infty f(x) dx$ converges and that $f$ is symmetric about the origin. What does $f$ look like? Is it possible to identify a parametric set of solutions for $f$? How do you even go about solving a problem like this?

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What you have there looks like an integral equation of convolution type... except what's supposed to be known in that case is also unknown. –  J. M. Jul 13 '12 at 7:53

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up vote 3 down vote accepted

A Fourier transformation leads to $\hat f^2=\hat f$, which is solved by any Fourier transform $\hat f$ that takes only the values $0$ and $1$. For instance, $\hat f=\chi_{[-a,a]}$, the characteristic function on an interval centred on the origin, leads to a $\operatorname{sinc}$-like function $f$.

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This is very helpful, and it gives $f$ some properties that I really hoped it would have. Thank you! –  GMB Jul 13 '12 at 16:04
    
@user1050699: You're welcome! –  joriki Jul 13 '12 at 16:20

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