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Fourier duality for fractal measures with affine scales

Authors: Dorin Ervin Dutkay and Palle E. T. Jorgensen
Journal: Math. Comp. 81 (2012), 2253-2273
MSC (2010): Primary 47B32, 42B05, 28A35, 26A33, 62L20
Published electronically: May 22, 2012
MathSciNet review: 2945155
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Abstract: For a family of fractal measures, we find an explicit Fourier duality. The measures in the pair have compact support in $ \mathbb{R}^d$, and they both have the same matrix scaling; but the two use different translation vectors, one by a subset $ B$ in $ \mathbb{R}^d$, and the other by a related subset $ L$. Among other things, we show that there is then a pair of infinite discrete sets $ \Gamma (L)$ and $ \Gamma (B)$ in $ \mathbb{R}^d$ such that the $ \Gamma (L)$-Fourier exponentials are orthogonal in $ L^2(\mu _B)$, and the $ \Gamma (B)$-Fourier exponentials are orthogonal in $ L^2(\mu _L)$. These sets of orthogonal ``frequencies'' are typically lacunary, and they will be obtained by scaling in the large. The nature of our duality is explored below both in higher dimensions and for examples on the real line.

Our duality pairs do not always yield orthonormal Fourier bases in the respective $ L^2(\mu )$-Hilbert spaces, but depending on the geometry of certain finite orbits, we show that they do in some cases. We further show that there are new and surprising scaling symmetries of relevance for the ergodic theory of these affine fractal measures.

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  • [BJ99] Ola Bratteli and Palle E. T. Jorgensen.
    Iterated function systems and permutation representations of the Cuntz algebra.
    Mem. Amer. Math. Soc., 139(663):x+89, 1999. MR 1469149 (99k:46094a)
  • [CHK97] R. Craigen, W. H. Holzmann, and H. Kharaghani.
    On the asymptotic existence of complex Hadamard matrices.
    J. Combin. Des., 5(5):319-327, 1997. MR 1465343 (99e:05026)
  • [CKS08] Wojciech Czaja, Gitta Kutyniok, and Darrin Speegle.
    Beurling dimension of Gabor pseudoframes for affine subspaces.
    J. Fourier Anal. Appl., 14(4):514-537, 2008. MR 2421575 (2009k:42062)
  • [Den09] Qi-Rong Deng.
    Reverse iterated function system and dimension of discrete fractals.
    Bull. Aust. Math. Soc., 79(1):37-47, 2009. MR 2486879 (2010j:28013)
  • [DFdGtHR04] Remco Duits, Luc Florack, Jan de Graaf, and Bart ter Haar Romeny.
    On the axioms of scale space theory.
    J. Math. Imaging Vision, 20(3):267-298, 2004. MR 2060148 (2005k:94005)
  • [DHS09] Dorin Ervin Dutkay, Deguang Han, and Qiyu Sun.
    On the spectra of a Cantor measure.
    Adv. Math., 221(1):251-276, 2009. MR 2509326 (2010f:28013)
  • [DHSW09] Dorin Ervin Dutkay, Deguang Han, Qiyu Sun, and Eric Weber.
    On the Beurling dimension of exponential frames
    Adv. Math. 226 (2011), no. 1. MR 2735759
  • [Dit03] P. Dita.
    On the parametrization of complex Hadamard matrices.
    Romanian J. Phys., 48(5-6):619-626 (2004), 2003. MR 2145440 (2006c:81035)
  • [Dit04] P. Dita.
    Some results on the parametrization of complex Hadamard matrices.
    J. Phys. A, 37(20):5355-5374, 2004. MR 2065675 (2005b:15045)
  • [DJ06a] Dorin E. Dutkay and Palle E. T. Jorgensen.
    Wavelets on fractals.
    Rev. Mat. Iberoam., 22(1):131-180, 2006. MR 2268116 (2008h:42071)
  • [DJ06b] Dorin Ervin Dutkay and Palle E. T. Jorgensen.
    Iterated function systems, Ruelle operators, and invariant projective measures.
    Math. Comp., 75(256):1931-1970 (electronic), 2006. MR 2240643 (2008h:28005)
  • [DJ07a] Dorin Ervin Dutkay and Palle E. T. Jorgensen.
    Analysis of orthogonality and of orbits in affine iterated function systems.
    Math. Z., 256(4):801-823, 2007. MR 2308892 (2009e:42013)
  • [DJ07b] Dorin Ervin Dutkay and Palle E. T. Jorgensen.
    Fourier frequencies in affine iterated function systems.
    J. Funct. Anal., 247(1):110-137, 2007. MR 2319756 (2008f:42007)
  • [DJ07c] Dorin Ervin Dutkay and Palle E. T. Jorgensen.
    Harmonic analysis and dynamics for affine iterated function systems.
    Houston J. Math., 33(3):877-905, 2007. MR 2335741 (2009k:42016)
  • [DJ07d] Dorin Ervin Dutkay and Palle E. T. Jorgensen.
    Martingales, endomorphisms, and covariant systems of operators in Hilbert space.
    J. Operator Theory, 58(2):269-310, 2007. MR 2358531 (2009h:47040)
  • [DJ08] Dorin Ervin Dutkay and Palle E. T. Jorgensen.
    Fourier series on fractals: a parallel with wavelet theory.
    In Radon transforms, geometry, and wavelets, volume 464 of Contemp. Math., pages 75-101. Amer. Math. Soc., Providence, RI, 2008. MR 2440130 (2010a:42138)
  • [DJ09] Dorin Ervin Dutkay and Palle E. T. Jorgensen.
    Quasiperiodic spectra and orthogonality for iterated function system measures.
    Math. Z., 261(2):373-397, 2009. MR 2457304 (2010b:28013)
  • [DvA08] Remco Duits and Markus van Almsick.
    The explicit solutions of linear left-invariant second order stochastic evolution equations on the 2D Euclidean motion group.
    Quart. Appl. Math., 66(1):27-67, 2008. MR 2396651 (2010c:37177)
  • [FLS09] Kevin Ford, Florian Luca, and Igor E. Shparlinski.
    On the largest prime factor of the Mersenne numbers.
    Bull. Aust. Math. Soc., 79(3):455-463, 2009. MR 2505350 (2010b:11126)
  • [Fug74] Bent Fuglede.
    Commuting self-adjoint partial differential operators and a group theoretic problem.
    J. Functional Analysis, 16:101-121, 1974. MR 0470754 (57:10500)
  • [GIL09] Daniele Guido, Tommaso Isola, and Michel L. Lapidus.
    A trace on fractal graphs and the Ihara zeta function.
    Trans. Amer. Math. Soc., 361(6):3041-3070, 2009. MR 2485417 (2010g:11148)
  • [HL08] Tian-You Hu and Ka-Sing Lau.
    Spectral property of the Bernoulli convolutions.
    Adv. Math., 219(2):554-567, 2008. MR 2435649 (2010a:42094)
  • [Hut81] John E. Hutchinson.
    Fractals and self-similarity.
    Indiana Univ. Math. J., 30(5):713-747, 1981. MR 625600 (82h:49026)
  • [IKP99] Alex Iosevich, Nets Katz, and Steen Pedersen.
    Fourier bases and a distance problem of Erdös.
    Math. Res. Lett., 6(2):251-255, 1999. MR 1689215 (2000j:42013)
  • [IKT03] Alex Iosevich, Nets Katz, and Terence Tao.
    The Fuglede spectral conjecture holds for convex planar domains.
    Math. Res. Lett., 10(5-6):559-569, 2003. MR 2024715 (2004i:42020)
  • [IW08] Marius Ionescu and Yasuo Watatani.
    $ C\sp \ast $-algebras associated with Mauldin-Williams graphs.
    Canad. Math. Bull., 51(4):545-560, 2008. MR 2462459 (2010a:46143)
  • [Jør82a] Palle E. T. Jørgensen.
    A generalization to locally compact abelian groups of a spectral problem for commuting partial differential operators.
    J. Pure Appl. Algebra, 25(3):297-301, 1982. MR 666022 (84h:43019)
  • [Jør82b] Palle E. T. Jørgensen.
    Spectral theory of finite volume domains in $ {\bf R}\sp {n}$.
    Adv. in Math., 44(2):105-120, 1982. MR 658536 (84k:47024)
  • [JP91] Palle E. T. Jorgensen and Steen Pedersen.
    An algebraic spectral problem for $ L\sp 2(\Omega ),\ \Omega \subset {\bf R}\sp n$.
    C. R. Acad. Sci. Paris Sér. I Math., 312(7):495-498, 1991. MR 1099679 (92b:47043)
  • [JP98] Palle E. T. Jorgensen and Steen Pedersen.
    Dense analytic subspaces in fractal $ L\sp 2$-spaces.
    J. Anal. Math., 75:185-228, 1998. MR 1655831 (2000a:46045)
  • [JP99] Palle E. T. Jorgensen and Steen Pedersen.
    Spectral pairs in Cartesian coordinates.
    J. Fourier Anal. Appl., 5(4):285-302, 1999. MR 1700084 (2002d:42027)
  • [ŁW02] Izabella Łaba and Yang Wang.
    On spectral Cantor measures.
    J. Funct. Anal., 193(2):409-420, 2002. MR 1929508 (2003g:28017)
  • [MP04] Leo Murata and Carl Pomerance.
    On the largest prime factor of a Mersenne number.
    In Number theory, volume 36 of CRM Proc. Lecture Notes, pages 209-218. Amer. Math. Soc., Providence, RI, 2004. MR 2076597 (2005i:11137)
  • [Mum94] David Mumford.
    Pattern theory: a unifying perspective.
    In First European Congress of Mathematics, Vol. I (Paris, 1992), volume 119 of Progr. Math., pages 187-224. Birkhäuser, Basel, 1994. MR 1341824
  • [Odl78] A. M. Odlyzko.
    Nonnegative digit sets in positional number systems.
    Proc. London Math. Soc. (3), 37(2):213-229, 1978. MR 507604 (80m:10004)
  • [Ped04] Steen Pedersen.
    On the dual spectral set conjecture.
    In Current trends in operator theory and its applications, volume 149 of Oper. Theory Adv. Appl., pages 487-491. Birkhäuser, Basel, 2004. MR 2063764 (2005h:42016)
  • [Str99] Robert S. Strichartz.
    Analysis on fractals.
    Notices Amer. Math. Soc., 46(10):1199-1208, 1999. MR 1715511 (2000i:58035)
  • [Str00] Robert S. Strichartz.
    Mock Fourier series and transforms associated with certain Cantor measures.
    J. Anal. Math., 81:209-238, 2000. MR 1785282 (2001i:42009)
  • [SW99] Robert S. Strichartz and Yang Wang.
    Geometry of self-affine tiles. I.
    Indiana Univ. Math. J., 48(1):1-23, 1999. MR 1722192 (2000k:52017)
  • [SZ08] Adam Skalski and Joachim Zacharias.
    Noncommutative topological entropy of endomorphisms of Cuntz algebras.
    Lett. Math. Phys., 86(2-3):115-134, 2008. MR 2465749 (2010c:46145)
  • [Tao04] Terence Tao.
    Fuglede's conjecture is false in 5 and higher dimensions.
    Math. Res. Lett., 11(2-3):251-258, 2004. MR 2067470 (2005i:42037)
  • [Vas06] D. V. Vasilev.
    The Lucas-Lehmer test for Mersenne numbers.
    Vestsī Nats. Akad. Navuk Belarusī Ser. Fīz.-Mat. Navuk, (2):113-115, 129, 2006. MR 2296562 (2008c:11010)
  • [XCQ05] Mingyuan Xia, Yingbao Chen, and Hong Qin.
    Some results for the existence of regular complex Hadamard matrices.
    Util. Math., 68:103-108, 2005. MR 2189697 (2006f:05030)

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Additional Information

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

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

Keywords: Affine fractal, Cantor set, Cantor measure, iterated function system, Hilbert space, Beurling density, Fourier bases.
Received by editor(s): November 5, 2009
Received by editor(s) in revised form: June 19, 2011
Published electronically: May 22, 2012
Additional Notes: This work was supported in part by the National Science Foundation.
Article copyright: © Copyright 2012 American Mathematical Society

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