Abstract

In this study of frames for measures on Cantor sets, we consider four measures with support contained in a Cantor set. These are the mass distribution measure, the Hausdorff measure of the appropriate dimension restricted to the Cantor set, the unique measure from Hutchinson's theorem for self-similar sets and the unique Lebesgue-Stieltjes measure with respect to the Cantor-Lebesgue function. For the ternary and quaternary Cantor sets, respectively, we show that
these four measures give the same measure $\mu$. This allows us to study frames in $L^2(\mu)$.

While the theory of frames is well developed, the literature on frames on Cantor
sets is recent and limited. Central in defining frames of exponentials on Cantor sets is the set of integers (hereafter called \textit{spectrum}) obtained from the Fourier transform of each measure supported on the corresponding Cantor set. After giving some background on frames, we follow the work of Jorgensen and
Pedersen (1998) to find the spectrum of the mass distribution measure on the quaternary Cantor set from its Fourier transform and show that we do have an orthonormal basis, which is a special case of a frame. We also present the result of Jorgensen and Pedersen (1998) for the mass distribution measure on the ternary Cantor set, that it is not possible to have a spectrum that yields an orthonormal set of exponentials. However, this leads to the question: can we show the existence of a frame from the spectrum of the mass distribution measure on the ternary Cantor set? Recent work (for example, Lev (2018), Picioroaga and Weber (2017)) study this question but it remains an open problem.