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.
You may be aware of this already, but a similar (if not the same) question is addressed in Rudin's Fourier Analysis on Groups. Material on the same subject can be found in Katznelson's Harmonic analysis. I think pdfs of both of these books are available free online.
The relevant theorem is (I think) that of Beurling and Helson. If I remember right, the answer for $A(\mathbb{R})$ (which corresponds to those $\mu$ which are absolutely continuous with respect to Lebesgue measure) is that any $\alpha$ satisfying $A(\mathbb{R})\circ \alpha\subseteq A(\mathbb{R})$ must be of the form $\alpha(x) = \alpha_0 + \alpha_1 x$ for some $\alpha_1,\alpha_2 \in \mathbb{R}$. A similar statement holds for so-called Extremal algebras (this is established in a paper of Kamowitz and Wortman).
Sorry if I'm barking up the wrong tree here!
