  Consider an equation of the form $f(x)=g(x^k)$, where $k>1$ and $f(x)$ is a
function in a given Carleman class of smooth functions. For each $k$, we
construct a Carleman-type class which contains all the smooth solutions $g(x)$
to such equations. We prove, under regularity assumptions, that if the original
Carleman class is quasianalytic, then so is the new class. The results admit an
extension to multivariate functions.
