Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: 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 $\mu$.

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The categories of this paper are: solv-int
36
The categories of this paper are: cmp-lg
32
The categories of this paper are: atom-ph
30
The categories of this paper are: q-alg
35
The categories of this paper are: mtrl-th
28
The categories of this paper are: alg-geom
9
The categories of this paper are: nlin
18
The categories of this paper are: econ
14
The categories of this paper are: math-ph
29
The categories of this paper are: funct-an
27
The categories of this paper are: comp-gas
10
The categories of this paper are: q-bio
21
The categories of this paper are: adap-org
34
The categories of this paper are: chem-ph
17
The categories of this paper are: q-fin
4
The categories of this paper are: cond-mat