Fourier bases on fractals
The study of Bernoulli convolution measures, which are supported on Cantor subsets of the real line, dates back to the 1930's, and have experienced a resurgence with the connection between the measures and iterated function systems. We will use this IFS approach to consider the question of Fourier bases on the L^2 spaces with respect to Bernoulli convolution measures.
There are some interesting phenomena that arise in this setting, which will be discussed in this presentation. We find that some Cantor sets support Fourier bases while others do not. In cases where a Fourier basis does exist, we can sometimes scale or shift the Fourier frequencies by an integer to obtain another ONB. We also discover properties of the unitary operator mapping between two such bases. The self-similarity of the measure and the support space can, in some cases, carry over into a self-similarity of the operator.