Given a function $f(x)$ and its Fourier transform $\tilde f(k)$, I know from the convolution theorem that the Fourier transform of $g(x):=\sqrt{f(x)}$ fulfils $$\tilde g(k)\ast\tilde g(k) = \tilde f(k)$$ up to some normalization factors. But now I need to find the inverse of auto-convolution. Does this help somehow or is it rather the other way around that the "convolution-squareroot" is obtained via the Fourier transform?
2026-03-25 09:25:23.1774430723
What's the Fourier transform of a function's square root?
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