Wavelets on fractals

Mon, 01/06/2015 - 14:00

The class of fractals referred to are those which may be specified by a finite system of affine transformations,
assuming contractive scaling; and their corresponding selfsimilar measures, $\mu$. They include standard Cantor
spaces such as the middle third, and the planar Sierpinski caskets in various forms, and their corresponding
selfsimilar measures, but the class is more general than this; including fractals realized in $\mathbb R^d$, for
$d > 2$.
In part 1, we motivate the need for wavelets in the harmonic analysis of these selfsimilar measures $\mu$. While
classes of the Hilbert spaces $L^2(\mu)$ have Fourier bases, it is known (the speaker and Pedersen) that many do
not, for example the middle third Cantor can have no more than two orthogonal Fourier frequencies.
In part 2 of the talk, we outline a construction by the speaker and Dutkay to the effect that all the affine systems
do have wavelet bases; this entails what we call thin Cantor spaces.