Stable recovery of fractal measures by polynomial sampling

Abstract The problem of using iterated function systems (IFS) to approximate fractal measures from polynomial samples is studied. A method based on generalized moments is proposed. This method does not suffer from the ill-conditioning of the classical moment sampling method and allows for more samples to be taken. Error bounds are derived for general polynomial sampling methods. We also discuss various applications, including thermodynamical properties of harmonic solids. One striking result is the computation of the zero point vibrational energy of a face centered cubic crystal. Although our method does not provide upper and lower bounds, constructing an IFS with 10 moments provides a comparable accuracy as the standard computation with 30 moments using Pade approximations (which are a special type of IFS).