Denote by $B(\mathbb{R})$ the set of all functions on $\mathbb{R}$ which are representable in the form $f(x)=\int_{\mathbb{R}}e^{itx}d\mu(t)$, where $\mu$ is a finite complex-valued Borel measure.

Question: Is there a description of all functions $\alpha:\mathbb{R}\to\mathbb{R}$ such that $f\circ\alpha\in B(\mathbb{R})$ for all $f\in B(\mathbb{R})$?

For example, $\alpha(x):=cx$, $c\in\mathbb{R}$ satisfies these conditions.

I'm sure there's an answer somewhere.