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Convolution square root of delta

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Convolution square root of delta

I want to determine all distributional solutions of the convolution equation

f*f = delta

where delta denotes the Dirac delta function, i.e. (delta, g) = g(0) for any smooth compactly supported test function g. Fourier transforming I obtain

F^2 = 1

where F is the Fourier transform of f. But can I give a better, more explicit, characterization of those functions satisfying f*f = delta?

What is A?

Something got screwed up. What I meant was F = 1 or -1.

Unless I'm missing something, don't you have F = ±1 ?

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