Fourier duality for fractal measures with affine scales
HTML articles powered by AMS MathViewer
- by Dorin Ervin Dutkay and Palle E. T. Jorgensen PDF
- Math. Comp. 81 (2012), 2253-2273 Request permission
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.
References
- Ola Bratteli and Palle E. T. Jorgensen, Iterated function systems and permutation representations of the Cuntz algebra, Mem. Amer. Math. Soc. 139 (1999), no. 663, x+89. MR 1469149, DOI 10.1090/memo/0663
- R. Craigen, W. H. Holzmann, and H. Kharaghani, On the asymptotic existence of complex Hadamard matrices, J. Combin. Des. 5 (1997), no. 5, 319–327. MR 1465343, DOI 10.1002/(SICI)1520-6610(1997)5:5<319::AID-JCD1>3.3.CO;2-W
- Wojciech Czaja, Gitta Kutyniok, and Darrin Speegle, Beurling dimension of Gabor pseudoframes for affine subspaces, J. Fourier Anal. Appl. 14 (2008), no. 4, 514–537. MR 2421575, DOI 10.1007/s00041-008-9026-0
- Qi-Rong Deng, Reverse iterated function system and dimension of discrete fractals, Bull. Aust. Math. Soc. 79 (2009), no. 1, 37–47. MR 2486879, DOI 10.1017/S000497270800097X
- Remco Duits, Luc Florack, Jan de Graaf, and Bart ter Haar Romeny, On the axioms of scale space theory, J. Math. Imaging Vision 20 (2004), no. 3, 267–298. MR 2060148, DOI 10.1023/B:JMIV.0000024043.96722.aa
- Dorin Ervin Dutkay, Deguang Han, and Qiyu Sun, On the spectra of a Cantor measure, Adv. Math. 221 (2009), no. 1, 251–276. MR 2509326, DOI 10.1016/j.aim.2008.12.007
- Dorin Ervin Dutkay, Deguang Han, Qiyu Sun, and Eric Weber, On the Beurling dimension of exponential frames, Adv. Math. 226 (2011), no. 1, 285–297. MR 2735759, DOI 10.1016/j.aim.2010.06.017
- P. Diţă, On the parametrization of complex Hadamard matrices, Romanian J. Phys. 48 (2003), no. 5-6, 619–626 (2004). MR 2145440
- P. Diţă, Some results on the parametrization of complex Hadamard matrices, J. Phys. A 37 (2004), no. 20, 5355–5374. MR 2065675, DOI 10.1088/0305-4470/37/20/008
- Dorin E. Dutkay and Palle E. T. Jorgensen, Wavelets on fractals, Rev. Mat. Iberoam. 22 (2006), no. 1, 131–180. MR 2268116, DOI 10.4171/RMI/452
- Dorin Ervin Dutkay and Palle E. T. Jorgensen, Iterated function systems, Ruelle operators, and invariant projective measures, Math. Comp. 75 (2006), no. 256, 1931–1970. MR 2240643, DOI 10.1090/S0025-5718-06-01861-8
- Dorin Ervin Dutkay and Palle E. T. Jorgensen, Analysis of orthogonality and of orbits in affine iterated function systems, Math. Z. 256 (2007), no. 4, 801–823. MR 2308892, DOI 10.1007/s00209-007-0104-9
- Dorin Ervin Dutkay and Palle E. T. Jorgensen, Fourier frequencies in affine iterated function systems, J. Funct. Anal. 247 (2007), no. 1, 110–137. MR 2319756, DOI 10.1016/j.jfa.2007.03.002
- Dorin Ervin Dutkay and Palle E. T. Jorgensen, Harmonic analysis and dynamics for affine iterated function systems, Houston J. Math. 33 (2007), no. 3, 877–905. MR 2335741
- Dorin Ervin Dutkay and Palle E. T. Jorgensen, Martingales, endomorphisms, and covariant systems of operators in Hilbert space, J. Operator Theory 58 (2007), no. 2, 269–310. MR 2358531
- Dorin Ervin Dutkay and Palle E. T. Jorgensen, Fourier series on fractals: a parallel with wavelet theory, Radon transforms, geometry, and wavelets, Contemp. Math., vol. 464, Amer. Math. Soc., Providence, RI, 2008, pp. 75–101. MR 2440130, DOI 10.1090/conm/464/09077
- Dorin Ervin Dutkay and Palle E. T. Jorgensen, Quasiperiodic spectra and orthogonality for iterated function system measures, Math. Z. 261 (2009), no. 2, 373–397. MR 2457304, DOI 10.1007/s00209-008-0329-2
- 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 (2008), no. 1, 27–67. MR 2396651, DOI 10.1090/S0033-569X-07-01066-0
- Kevin Ford, Florian Luca, and Igor E. Shparlinski, On the largest prime factor of the Mersenne numbers, Bull. Aust. Math. Soc. 79 (2009), no. 3, 455–463. MR 2505350, DOI 10.1017/S0004972709000033
- Bent Fuglede, Commuting self-adjoint partial differential operators and a group theoretic problem, J. Functional Analysis 16 (1974), 101–121. MR 0470754, DOI 10.1016/0022-1236(74)90072-x
- Daniele Guido, Tommaso Isola, and Michel L. Lapidus, A trace on fractal graphs and the Ihara zeta function, Trans. Amer. Math. Soc. 361 (2009), no. 6, 3041–3070. MR 2485417, DOI 10.1090/S0002-9947-08-04702-8
- Tian-You Hu and Ka-Sing Lau, Spectral property of the Bernoulli convolutions, Adv. Math. 219 (2008), no. 2, 554–567. MR 2435649, DOI 10.1016/j.aim.2008.05.004
- John E. Hutchinson, Fractals and self-similarity, Indiana Univ. Math. J. 30 (1981), no. 5, 713–747. MR 625600, DOI 10.1512/iumj.1981.30.30055
- Alex Iosevich, Nets Katz, and Steen Pedersen, Fourier bases and a distance problem of Erdős, Math. Res. Lett. 6 (1999), no. 2, 251–255. MR 1689215, DOI 10.4310/MRL.1999.v6.n2.a13
- Alex Iosevich, Nets Katz, and Terence Tao, The Fuglede spectral conjecture holds for convex planar domains, Math. Res. Lett. 10 (2003), no. 5-6, 559–569. MR 2024715, DOI 10.4310/MRL.2003.v10.n5.a1
- Marius Ionescu and Yasuo Watatani, $C^\ast$-algebras associated with Mauldin-Williams graphs, Canad. Math. Bull. 51 (2008), no. 4, 545–560. MR 2462459, DOI 10.4153/CMB-2008-054-0
- 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 (1982), no. 3, 297–301. MR 666022, DOI 10.1016/0022-4049(82)90084-6
- Palle E. T. Jørgensen, Spectral theory of finite volume domains in $\textbf {R}^{n}$, Adv. in Math. 44 (1982), no. 2, 105–120. MR 658536, DOI 10.1016/0001-8708(82)90001-9
- Palle E. T. Jorgensen and Steen Pedersen, An algebraic spectral problem for $L^2(\Omega ),\ \Omega \subset \textbf {R}^n$, C. R. Acad. Sci. Paris Sér. I Math. 312 (1991), no. 7, 495–498 (English, with French summary). MR 1099679
- Palle E. T. Jorgensen and Steen Pedersen, Dense analytic subspaces in fractal $L^2$-spaces, J. Anal. Math. 75 (1998), 185–228. MR 1655831, DOI 10.1007/BF02788699
- Palle E. T. Jorgensen and Steen Pedersen, Spectral pairs in Cartesian coordinates, J. Fourier Anal. Appl. 5 (1999), no. 4, 285–302. MR 1700084, DOI 10.1007/BF01259371
- Izabella Łaba and Yang Wang, On spectral Cantor measures, J. Funct. Anal. 193 (2002), no. 2, 409–420. MR 1929508, DOI 10.1006/jfan.2001.3941
- Leo Murata and Carl Pomerance, On the largest prime factor of a Mersenne number, Number theory, CRM Proc. Lecture Notes, vol. 36, Amer. Math. Soc., Providence, RI, 2004, pp. 209–218. MR 2076597, DOI 10.1090/crmp/036/16
- David Mumford, Pattern theory: a unifying perspective, First European Congress of Mathematics, Vol. I (Paris, 1992) Progr. Math., vol. 119, Birkhäuser, Basel, 1994, pp. 187–224. MR 1341824
- A. M. Odlyzko, Nonnegative digit sets in positional number systems, Proc. London Math. Soc. (3) 37 (1978), no. 2, 213–229. MR 507604, DOI 10.1112/plms/s3-37.2.213
- Steen Pedersen, On the dual spectral set conjecture, Current trends in operator theory and its applications, Oper. Theory Adv. Appl., vol. 149, Birkhäuser, Basel, 2004, pp. 487–491. MR 2063764
- Robert S. Strichartz, Analysis on fractals, Notices Amer. Math. Soc. 46 (1999), no. 10, 1199–1208. MR 1715511
- Robert S. Strichartz, Mock Fourier series and transforms associated with certain Cantor measures, J. Anal. Math. 81 (2000), 209–238. MR 1785282, DOI 10.1007/BF02788990
- Robert S. Strichartz and Yang Wang, Geometry of self-affine tiles. I, Indiana Univ. Math. J. 48 (1999), no. 1, 1–23. MR 1722192, DOI 10.1512/iumj.1999.48.1616
- Adam Skalski and Joachim Zacharias, Noncommutative topological entropy of endomorphisms of Cuntz algebras, Lett. Math. Phys. 86 (2008), no. 2-3, 115–134. MR 2465749, DOI 10.1007/s11005-008-0279-y
- Terence Tao, Fuglede’s conjecture is false in 5 and higher dimensions, Math. Res. Lett. 11 (2004), no. 2-3, 251–258. MR 2067470, DOI 10.4310/MRL.2004.v11.n2.a8
- D. V. Vasil′ev, The Lucas-Lehmer test for Mersenne numbers, Vestsī Nats. Akad. Navuk Belarusī Ser. Fīz.-Mat. Navuk 2 (2006), 113–115, 129 (Russian, with English and Russian summaries). MR 2296562
- Mingyuan Xia, Yingbao Chen, and Hong Qin, Some results for the existence of regular complex Hadamard matrices, Util. Math. 68 (2005), 103–108. MR 2189697
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
- MR Author ID: 608228
- Email: ddutkay@mail.ucf.edu
- Palle E. T. Jorgensen
- Affiliation: University of Iowa, Department of Mathematics, 14 MacLean Hall, Iowa City, Iowa 52242-1419
- MR Author ID: 95800
- ORCID: 0000-0003-2681-5753
- Email: jorgen@math.uiowa.edu
- 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.
- © Copyright 2012 American Mathematical Society
- Journal: Math. Comp. 81 (2012), 2253-2273
- MSC (2010): Primary 47B32, 42B05, 28A35, 26A33, 62L20
- DOI: https://doi.org/10.1090/S0025-5718-2012-02580-4
- MathSciNet review: 2945155