Polynomial approximation of self-similar measures and the spectrum of the transfer operator

We consider self-similar measures on $\mathbb{R}\mathrm{.}$ The Hutchinson operator $H$ acts on measures and is the dual of the transfer operator $T$ which acts on continuous functions. We determine polynomial eigenfunctions of $T\mathrm{.}$ As a consequence, we obtain eigenvalues of $H$ and natural polynomial approximations of the self-similar measure. Bernoulli convolutions are studied as an example.