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Affine fractals as boundaries and their harmonic analysis

Authors: Dorin Ervin Dutkay and Palle E. T. Jorgensen
Journal: Proc. Amer. Math. Soc. 139 (2011), 3291-3305
MSC (2010): Primary 47B32, 42B05, 28A35, 26A33, 62L20
Published electronically: February 8, 2011
MathSciNet review: 2811284
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Abstract: 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(\mathbb{D})$, $ \mathbb{D}$ denoting the disk, such that for every function $ f$ in $ H^2(\Gamma)$ and for every point $ z$ in $ \mathbb{D}$, $ f(z)$ admits a boundary integral represented by an associated measure $ \mu$, with integration over $ {\textrm{supp}}(\mu)$ placed as a Cantor subset on the circle $ \mathbb{T} :=$   bd$ (\mathbb{D})$.

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, missing parts, or gaps. Given $ \Gamma$, we find corresponding generalized Szegö kernels $ G_\Gamma$, and we compare them to the classical Szegö kernel for $ \mathbb{D}$.

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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Additional Information

Dorin Ervin Dutkay
Affiliation: Department of Mathematics, University of Central Florida, 4000 Central Florida Boulevard, P.O. Box 161364, Orlando, Florida 32816-1364

Palle E. T. Jorgensen
Affiliation: Department of Mathematics, 14 MacLean Hall, University of Iowa, Iowa City, Iowa 52242-1419

Keywords: Affine fractal, Cantor set, Cantor measure, iterated function system, Hilbert space, Fourier bases.
Received by editor(s): June 2, 2010
Received by editor(s) in revised form: August 21, 2010, and August 23, 2010
Published electronically: February 8, 2011
Additional Notes: This work was supported in part by the National Science Foundation.
Communicated by: Nigel J. Kalton
Article copyright: © Copyright 2011 American Mathematical Society