Affine Iterated Function Systems invariant measures and their approximation Iteriertes Funktionensystem invariantes Maß Hutchinson Operator Transfer Operator Bernoulli Faltungen Spektrum Iterated Function System invariant measure Hutchinson Operator Transfer Operator Bernoulli convolutions spectrum We consider Iterated Function Systems (IFS) on the real line and on the complex plane. Every IFS defines a self similar measure supported on a self similar set. We study the transfer operator (which acts on the space of continuous functions on the self similar set) and the Hutchinson operator (which acts on the space of Borel regular measures on the self similar set). We show that the transfer operator has an infinitely countable set of polynomial eigenfunctions. These eigenfunctions can be regarded as generalized Bernoulli polynomials. The polynomial eigenfuctions define a polynomial approximation of the self similar measure. We also study the moments of the self similar measure and give recursions for computing them. Further we develop a numerical method based on Markov chains to study the spectrum of the Hutchinson and transfer operators. This method provides numerical approximations of the invariant measure for which we give error bounds in terms of the Wasserstein distance. The standard example in this thesis is the parametric family of Bernoulli convolutions.