What are the continuous functions $f$ such that on $\mathbb{R}^{+*}$, they satisfy following functional equation:

$$\int_0^\infty f(t) e^{- i t x} dt =\lambda \frac{1}{x} f(\frac{1}{x})$$

$\lambda$ is a constant.

The functions $f(x)=x^{\alpha}$ with $-1<Re(\alpha)<0$ are solution, but can we find other solutions to this equation ? Any method to solve this problem ? I tried to transform it to find a differential equation but did not succeed ...

What are the continuous functions $f$ such that on $\mathbb{R}^{+*}$, they satisfy following functional equation:

$$\int_0^\infty f(t) e^{-itx} \, dt =\lambda \frac{1}{x} f\left(\frac{1}{x}\right)$$

$\lambda$ is a constant.

The functions $f(x)=x^{\alpha}$ with $-1<\operatorname{Re}(\alpha)<0$ are solution, but can we find other solutions to this equation ? Any method to solve this problem ? I tried to transform it to find a differential equation but did not succeed ...