Affine fractals as boundaries and their harmonic analysis

We introduce the notion of boundary representation for fractal Fourier expansions, starting with a familiar notion of spectral pairs for affine fractal measures. Specializing to one dimension, we establish boundary representations for these fractals. We prove that as sets these fractals arise as boundaries of functions in closed subspaces of the Hardy space $H^2$. By this we mean that there are lacunary subsets $\Gamma$ of the non-negative integers, and associated closed $\Gamma$-subspace in the Hardy space $H^2(\bd)$, $\bd$ denoting the disk, such that for every function $f$ in in $H^2(\Gamma)$, and for every point $z$ in $\bd$, $f(z)$ admits a boundary integral represented by an associated measure $\mu$, with integration over $\supp{\mu}$ placed as a Cantor subset on the circle $\bt := \{bd}(\bd)$. We study families of pairs: measures $\mu$ and sets $\Gamma$ of lacunary form, admitting lacunary Fourier series in $L^2(\mu)$; i.e., configurations $\Gamma$ arranged with a geometric progression of empty spacing, or missing parts, gaps. Given $\Gamma$, we find corresponding generalized Szeg\" o kernels $G_\Gamma$, and we compare them to the classical Szeg\" o kernel for $\bd$. Rather than the more traditional approach of starting with $\mu$, and then asking for possibilities for sets $\Gamma$, such that we get Fourier series representations, we turn the problem upside down; now starting instead with a countably infinite discrete subset $\Gamma$, and, within a new duality framework, we study the possibilities for choices of measures $\mu$.