Affine fractals as boundaries and their harmonic analysis
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- by Dorin Ervin Dutkay and Palle E. T. Jorgensen PDF
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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 $\mathbb {T} := \mbox {bd}(\mathbb {D})$.
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
- MR Author ID: 608228
- Email: ddutkay@mail.ucf.edu
- Palle E. T. Jorgensen
- Affiliation: Department of Mathematics, 14 MacLean Hall, University of Iowa, Iowa City, Iowa 52242-1419
- MR Author ID: 95800
- ORCID: 0000-0003-2681-5753
- Email: jorgen@math.uiowa.edu
- 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
- © Copyright 2011 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 139 (2011), 3291-3305
- MSC (2010): Primary 47B32, 42B05, 28A35, 26A33, 62L20
- DOI: https://doi.org/10.1090/S0002-9939-2011-10752-4
- MathSciNet review: 2811284