  Given a $\mathcal{C}^\infty$ expanding map $T$ of the circle, we construct a
Hilbert space $\mathcal{H}$ of smooth functions on which the transfer operator
$\mathcal{L}$ associated to $T$ acts as a compact operator. This result is made
quantitative (in terms of singular values of the operator $\mathcal{L}$ acting
on $\mathcal{H}$) using the language of Denjoy-Carleman classes. Moreover, the
nuclear power decomposition of Baladi and Tsujii can be performed on the space
$\mathcal{H}$, providing a bound on the growth of the dynamical determinant
associated to $\mathcal{L}$.
