Spectral measures and Cuntz algebras

Dutkay, Dorin; Jorgensen, Palle

doi:10.1090/S0025-5718-2012-02589-0

Spectral measures and Cuntz algebras
HTML articles powered by AMS MathViewer

by Dorin Ervin Dutkay and Palle E. T. Jorgensen;

Math. Comp. 81 (2012), 2275-2301

DOI: https://doi.org/10.1090/S0025-5718-2012-02589-0

Published electronically: February 14, 2012

PDF | Request permission

Abstract:

We consider a family of measures $\mu$ supported in $\mathbb {R}^d$ and generated in the sense of Hutchinson by a finite family of affine transformations. It is known that interesting sub-families of these measures allow for an orthogonal basis in $L^2(\mu )$ consisting of complex exponentials, i.e., a Fourier basis corresponding to a discrete subset $\Gamma$ in $\mathbb {R}^d$. Here we offer two computational devices for understanding the interplay between the possibilities for such sets $\Gamma$ (spectrum) and the measures $\mu$ themselves. Our computations combine the following three tools: duality, discrete harmonic analysis, and dynamical systems based on representations of the Cuntz $C^*$-algebras $\mathcal O_N$.

References

George E. Andrews, Richard Askey, and Ranjan Roy, Special functions, Encyclopedia of Mathematics and its Applications, vol. 71, Cambridge University Press, Cambridge, 1999. MR 1688958, DOI 10.1017/CBO9781107325937
Akram Aldroubi, Carlos Cabrelli, Douglas Hardin, and Ursula Molter, Optimal shift invariant spaces and their Parseval frame generators, Appl. Comput. Harmon. Anal. 23 (2007), no. 2, 273–283. MR 2344616, DOI 10.1016/j.acha.2007.05.001
K. T. Arasu, Warwick de Launey, and S. L. Ma, On circulant complex Hadamard matrices, Des. Codes Cryptogr. 25 (2002), no. 2, 123–142. MR 1883962, DOI 10.1023/A:1013817013980
Michael F. Barnsley, Transformations between self-referential sets, Amer. Math. Monthly 116 (2009), no. 4, 291–304. MR 2503315, DOI 10.4169/193009709X470155
Michael Barnsley, John Hutchinson, and Örjan Stenflo, A fractal valued random iteration algorithm and fractal hierarchy, Fractals 13 (2005), no. 2, 111–146. MR 2151094, DOI 10.1142/S0218348X05002799
O. Bratteli and P. E. T. Jorgensen, Endomorphisms of ${\scr B}({\scr H})$. II. Finitely correlated states on ${\scr O}_n$, J. Funct. Anal. 145 (1997), no. 2, 323–373. MR 1444086, DOI 10.1006/jfan.1996.3033
Ola Bratteli and Palle E. T. Jorgensen, Isometries, shifts, Cuntz algebras and multiresolution wavelet analysis of scale $N$, Integral Equations Operator Theory 28 (1997), no. 4, 382–443. MR 1465320, DOI 10.1007/BF01309155
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
O. Bratteli, P. E. T. Jorgensen, A. Kishimoto, and R. F. Werner, Pure states on $\scr O_d$, J. Operator Theory 43 (2000), no. 1, 97–143. MR 1740897
William D. Banks and Florian Luca, Sums of prime divisors and Mersenne numbers, Houston J. Math. 33 (2007), no. 2, 403–413. MR 2308986
Joseph A. Ball and Victor Vinnikov, Functional models for representations of the Cuntz algebra, Operator theory, systems theory and scattering theory: multidimensional generalizations, Oper. Theory Adv. Appl., vol. 157, Birkhäuser, Basel, 2005, pp. 1–60. MR 2129642, DOI 10.1007/3-7643-7303-2_{1}
D. Cerveau, J.-P. Conze, and A. Raugi, Ensembles invariants pour un opérateur de transfert dans $\textbf {R}^d$, Bol. Soc. Brasil. Mat. (N.S.) 27 (1996), no. 2, 161–186 (French, with English and French summaries). MR 1418931, DOI 10.1007/BF01259358
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
J.-P. Conze, L. Hervé, and A. Raugi, Pavages auto-affines, opérateurs de transfert et critères de réseau dans $\textbf {R}^d$, Bol. Soc. Brasil. Mat. (N.S.) 28 (1997), no. 1, 1–42 (French, with English and French summaries). MR 1444447, DOI 10.1007/BF01235987
Alain Connes and Matilde Marcolli, Renormalization, the Riemann-Hilbert correspondence, and motivic Galois theory, Frontiers in number theory, physics, and geometry. II, Springer, Berlin, 2007, pp. 617–713. MR 2290770, DOI 10.1007/978-3-540-30308-4_{1}3
Jean-Pierre Conze and Albert Raugi, Fonctions harmoniques pour un opérateur de transition et applications, Bull. Soc. Math. France 118 (1990), no. 3, 273–310 (French, with English summary). MR 1078079, DOI 10.24033/bsmf.2148
Joachim Cuntz, Simple $C^*$-algebras generated by isometries, Comm. Math. Phys. 57 (1977), no. 2, 173–185. MR 467330, DOI 10.1007/BF01625776
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
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, Palle E. T. Jorgensen, and Gabriel Picioroaga, Unitary representations of wavelet groups and encoding of iterated function systems in solenoids, Ergodic Theory Dynam. Systems 29 (2009), no. 6, 1815–1852. MR 2563094, DOI 10.1017/S0143385708000904
Xiaoyan Deng, Helong Li, and Xiaokun Chen, The symbol series expression and Hölder exponent estimates of fractal interpolation function, J. Comput. Anal. Appl. 11 (2009), no. 3, 507–523. MR 2530679
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 470754, DOI 10.1016/0022-1236(74)90072-x
Daniel Gonçalves and Danilo Royer, Perron-Frobenius operators and representations of the Cuntz-Krieger algebras for infinite matrices, J. Math. Anal. Appl. 351 (2009), no. 2, 811–818. MR 2473986, DOI 10.1016/j.jmaa.2008.11.018
Xing-Gang He and Ka-Sing Lau, On a generalized dimension of self-affine fractals, Math. Nachr. 281 (2008), no. 8, 1142–1158. MR 2427166, DOI 10.1002/mana.200510666
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
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, Analysis and probability: wavelets, signals, fractals, Graduate Texts in Mathematics, vol. 234, Springer, New York, 2006. MR 2254502
H. Kharaghani and Jennifer Seberry, The excess of complex Hadamard matrices, Graphs Combin. 9 (1993), no. 1, 47–56. MR 1215584, DOI 10.1007/BF01195326
King-Shun Leung and Ka-Sing Lau, Disklikeness of planar self-affine tiles, Trans. Amer. Math. Soc. 359 (2007), no. 7, 3337–3355. MR 2299458, DOI 10.1090/S0002-9947-07-04106-2
Alexandru Mihail and Radu Miculescu, The shift space for an infinite iterated function system, Math. Rep. (Bucur.) 11(61) (2009), no. 1, 21–32. MR 2506506
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
Ursula M. Molter and Leandro Zuberman, A fractal Plancherel theorem, Real Anal. Exchange 34 (2009), no. 1, 69–85. MR 2527123, DOI 10.14321/realanalexch.34.1.0069
Kasso A. Okoudjou and Robert S. Strichartz, Weak uncertainty principles on fractals, J. Fourier Anal. Appl. 11 (2005), no. 3, 315–331. MR 2167172, DOI 10.1007/s00041-005-4032-y
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
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
Klaus Thomsen, On the structure of beta shifts, Algebraic and topological dynamics, Contemp. Math., vol. 385, Amer. Math. Soc., Providence, RI, 2005, pp. 321–332. MR 2180243, DOI 10.1090/conm/385/07204
Zu-Guo Yu, Vo Anh, Ka-Sing Lau, and Ka-Hou Chu, The genomic tree of living organisms based on a fractal model, Phys. Lett. A 317 (2003), no. 3-4, 293–302. MR 2018655, DOI 10.1016/j.physleta.2003.08.040
Haibiao Zheng, Yanren Hou, Feng Shi, and Lina Song, A finite element variational multiscale method for incompressible flows based on two local Gauss integrations, J. Comput. Phys. 228 (2009), no. 16, 5961–5977. MR 2542923, DOI 10.1016/j.jcp.2009.05.006