Defining the operator $\phi$ by: $\phi(f(x))=\frac{1}{|x|} f(\frac{1}{x})$, and noting $\mathcal{F}$ the Fourier transform on the real line, can we characterize all the functions (with real variable and complex value) satisfying the following relation ?
$$\phi \circ \mathcal{F} (f(x))= c \; \mathcal{F} \circ \phi (f(x))$$
where $c$ is a complex constant.
All functions (tempered distribution to be precise) $f(x)=|x|^{\alpha}$ with $0>\Re (\alpha)>-1$, satisfy above equations but can we find other functions ?
The Second question with same operator $\phi$:
Can we find a way to characterize all functions $f$ satisfying
$$\int_0^{\infty}\phi \circ \mathcal{F} (f(x)) dx = 0$$
