Estimates on the Spectrum of Fractals Arising From Affine Iterations

In the first section we review recent results on the harmonic analysis of fractals generated by iterated function systems with emphasis on spectral duality. Classical harmonic analysis is typically based on groups whereas the fractals are most often not groups. We show that nonetheless those fractals that come from iteration of affine mappings in R^d have a spectral duality which is instead based on approximation and a certain dual affine system on the Fourier transform side. The present work is based on iteration of frame estimates (which have been studied earlier for regions in R^d). Our emphasis is on new results regarding the interplay between the limit-fractal on the one hand, and on the other the corresponding regions in R^d which generate iterated function systems of contractive affine mappings. As an application of our frame results, we obtain a classification of a certain type of spectral pairs.