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

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