Suppose we have a function $f: \mathbb{R} \rightarrow \mathbb{R}$ whose Fourier transform exists and is given by $\hat{f}$. Then the convolution theorem tells us that for any $k \in \mathbb{N}$, we have:
$$\displaystyle \widehat{f^k} = \underbrace{\hat{f} \ast ... \ast \hat{f}}_{k \text{ terms}},$$
where $f \ast g$ denotes the convolution of two functions on $\mathbb{R}.$ However, this does not tell us anything about non-integer values of $k$. For instance, suppose we could prove the existence of $\widehat{f^{1/2}}$. Is there any way in which we can compute the Fourier transform of $f^{1/2}$ explicitly, given only an explicit form for $\hat{f}$?