Affine fractals as boundaries and their harmonic analysis
Authors:
Dorin Ervin Dutkay and Palle E. T. Jorgensen
Journal:
Proc. Amer. Math. Soc. 139 (2011), 32913305
MSC (2010):
Primary 47B32, 42B05, 28A35, 26A33, 62L20
Published electronically:
February 8, 2011
MathSciNet review:
2811284
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Abstract 
References 
Similar Articles 
Additional Information
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 . By this we mean that there are lacunary subsets of the nonnegative integers and associated closed subspace in the Hardy space , denoting the disk, such that for every function in and for every point in , admits a boundary integral represented by an associated measure , with integration over placed as a Cantor subset on the circle bd. We study families of pairs: measures and sets of lacunary form, admitting lacunary Fourier series in ; i.e., configurations arranged with a geometric progression of empty spacing, missing parts, or gaps. Given , we find corresponding generalized Szegö kernels , and we compare them to the classical Szegö kernel for . Rather than the more traditional approach of starting with and then asking for possibilities for sets , such that we get Fourier series representations, we turn the problem upside down; now starting instead with a countably infinite discrete subset and within a new duality framework, we study the possibilities for choices of measures .
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Dorin Ervin Dutkay, Deguang Han, Qiyu Sun, and Eric Weber.
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 [ADV09]
 Daniel Alpay, Aad Dijksma, and Dan Volok.
Schur multipliers and de BrangesRovnyak spaces: the multiscale case. J. Operator Theory, 61(1):87118, 2009. MR 2481805 (2010f:46046)
 [AL08]
 Daniel Alpay and David Levanony.
On the reproducing kernel Hilbert spaces associated with the fractional and bifractional Brownian motions. Potential Anal., 28(2):163184, 2008. MR 2373103 (2009g:46043)
 [Aro50]
 N. Aronszajn.
Theory of reproducing kernels. Trans. Amer. Math. Soc., 68:337404, 1950. MR 0051437 (14:479c)
 [Arv98]
 William Arveson.
Subalgebras of algebras. III. Multivariable operator theory. Acta Math., 181(2):159228, 1998. MR 1668582 (2000e:47013)
 [CC08]
 Peter G. Casazza and Ole Christensen.
The reconstruction property in Banach spaces and a perturbation theorem. Canad. Math. Bull., 51(3):348358, 2008. MR 2436925 (2009g:42053)
 [CF09]
 Peter G. Casazza and Matthew Fickus.
Minimizing fusion frame potential. Acta Appl. Math., 107(13):724, 2009. MR 2520007 (2010e:42042)
 [CW08]
 Peter G. Casazza and Eric Weber.
The KadisonSinger problem and the uncertainty principle. Proc. Amer. Math. Soc., 136(12):42354243, 2008. MR 2431036 (2009m:42052)
 [DHS09]
 Dorin Ervin Dutkay, Deguang Han, and Qiyu Sun.
On the spectra of a Cantor measure. Adv. Math., 221(1):251276, 2009. MR 2509326 (2010f:28013)
 [DHSW10a]
 Dorin Ervin Dutkay, Deguang Han, Qiyu Sun, and Eric Weber.
Bessel sequences of exponentials on fractal measures. Preprint, 2010.
 [DHSW10b]
 Dorin Ervin Dutkay, Deguang Han, Qiyu Sun, and Eric Weber.
On the Beurling dimension of exponential frames. To appear in Adv. Math., 2010.
 [DJ06]
 Dorin Ervin Dutkay and Palle E. T. Jorgensen.
Iterated function systems, Ruelle operators, and invariant projective measures. Math. Comp., 75(256):19311970 (electronic), 2006. MR 2240643 (2008h:28005)
 [Hut81]
 John E. Hutchinson.
Fractals and selfsimilarity. Indiana Univ. Math. J., 30(5):713747, 1981. MR 625600 (82h:49026)
 [JP98]
 Palle E. T. Jorgensen and Steen Pedersen. Dense analytic subspaces in fractal spaces. J. Anal. Math., 75:185228, 1998. MR 1655831 (2000a:46045)
 [KS59]
 Richard V. Kadison and I. M. Singer.
Extensions of pure states. Amer. J. Math., 81:383400, 1959. MR 0123922 (23:A1243)
 [Lon67]
 Calvin T. Long.
Addition theorems for sets of integers. Pacific J. Math., 23:107112, 1967. MR 0215807 (35:6642)
 [ŁW02]
 Izabella Łaba and Yang Wang.
On spectral Cantor measures. J. Funct. Anal., 193(2):409420, 2002. MR 1929508 (2003g:28017)
 [Rud87]
 Walter Rudin.
Real and complex analysis. McGrawHill Book Co., New York, third edition, 1987. MR 924157 (88k:00002)
 [Str98]
 Robert S. Strichartz. Remarks on: ``Dense analytic subspaces in fractal spaces'' [J. Anal. Math. 75 (1998), 185228; MR1655831 (2000a:46045)] by P. E. T. Jorgensen and S. Pedersen. J. Anal. Math., 75:229231, 1998. MR 1655832 (2000a:46046)
 [Str00]
 Robert S. Strichartz.
Mock Fourier series and transforms associated with certain Cantor measures. J. Anal. Math., 81:209238, 2000. MR 1785282 (2001i:42009)
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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 328161364
Email:
ddutkay@mail.ucf.edu
Palle E. T. Jorgensen
Affiliation:
Department of Mathematics, 14 MacLean Hall, University of Iowa, Iowa City, Iowa 522421419
Email:
jorgen@math.uiowa.edu
DOI:
http://dx.doi.org/10.1090/S000299392011107524
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
