What are the continuous functions $f$ such that on $\mathbb{R}^{+*}$:

$$f(x) - \frac{C}{x} \hat{f}(\frac{1}{x}) =x^{\alpha}$$

Where $\hat{f}$ is the Fourier transform of $f(|x|)$ and $C$ a constant.

The functions $f(t)=t^{\alpha}$ with $-1<Re(\alpha)<0$ are solution, but can we find other solutions to this equation ?

For example solutions which have asymptotics $t^{\alpha}$ near zero (with $-1<Re(\alpha)<0$) and decrease rapidly to zero at infinty have a chance to exist, as in this case we have $\frac{C}{x} \hat{f}(\frac{1}{x})$ which is asymptotic to $t^{\alpha}$ at infinity and decreases to zero in zero (providing $\int_0^{\infty} f(t) dt=0$)