Hostname: page-component-78c5997874-g7gxr Total loading time: 0 Render date: 2024-11-05T16:59:18.635Z Has data issue: false hasContentIssue false
  • English
  • Français

Spectrality of self-affine measures on the three-dimensional Sierpinski gasket

Part of: Normed linear spaces and Banach spaces; Banach lattices Classical measure theory Nontrigonometric harmonic analysis

Published online by Cambridge University Press:  20 April 2012

Jian-Lin Li
Jian-Lin Li
Affiliation:
College of Mathematics and Information Science, Shaanxi Normal University, Xi'an 710062, People's Republic of China ([email protected])
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The self-affine measure μM, D corresponding to M = diag[p1, p2, p3] (pj ∈ ℤ \ {0, ± 1}, j = 1, 2, 3) and D = {0, e1, e2, e3} in the space ℝ3 is supported on the three-dimensional Sierpinski gasket T(M, D), where e1, e2, e3 are the standard basis of unit column vectors in ℝ3. We shall determine the spectrality and non-spectrality of μM, D, and show that if pj ∈ 2ℤ \ {0, 2} for j = 1, 2, 3, then μM, D is a spectral measure, and if pj ∈ (2ℤ + 1) \ {±1} for j = 1, 2, 3, then μM, D is a non-spectral measure and there exist at most 4 mutually orthogonal exponential functions in L2(μM, D), where the number 4 is the best possible. This generalizes the known results on the spectrality of self-affine measures.

Keywords

iterated function systemself-affine measureorthogonal exponentialsspectrality

MSC classification

Secondary: 28A80: Fractals 42C05: Orthogonal functions and polynomials, general theory 46C05: Hilbert and pre-Hilbert spaces
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 55 , Issue 2 , June 2012 , pp. 477 - 496
Copyright
Copyright © Edinburgh Mathematical Society 2012

References

1.Dutkay, D. E., Han, D. and Jorgensen, P. E. T., Orthogonal exponentials, translations, and Bohr completions, J. Funct. Analysis 257 (2009), 2999–3019.CrossRefGoogle Scholar
2.Dutkay, D. E., Han, D. and Sun, Q., On the spectra of a Cantor measure, Adv. Math. 221 (2009), 251–276.CrossRefGoogle Scholar
3.Dutkay, D. E. and Jorgensen, P. E. T., Iterated function systems, Rulle operators, and invariant projetive measures, Math. Comp. 75 (2006), 1931–1970.CrossRefGoogle Scholar
4.Dutkay, D. E. and Jorgensen, P. E. T., Analysis of orthogonality and of orbits in affine iterated function systems, Math. Z. 256 (2007), 801–823.CrossRefGoogle Scholar
5.Dutkay, D. E. and Jorgensen, P. E. T., Fourier frequencies in affine iterated function systems, J. Funct. Analysis 247 (2007), 110–137.CrossRefGoogle Scholar
6.Dutkay, D. E. and Jorgensen, P. E. T., Fourier series on fractals: a parallel with wavelet theory, in Radon transform, geometry, and wavelets, Contemporary Mathematics, Volume 464, pp. 75–101 (American Mathematical Society, 2008).CrossRefGoogle Scholar
7.Dutkay, D. E. and Jorgensen, P. E. T., Probability and Fourier duality for affine iterated function systems, Acta Appl. Math. 107 (2009), 293–311.CrossRefGoogle Scholar
8.Dutkay, D. E. and Jorgensen, P. E. T., Duality questions for operators, spectrum and measures, Acta Appl. Math. 108 (2009), 515–528.CrossRefGoogle Scholar
9.Dutkay, D. E. and Jorgensen, P. E. T., Quasiperiodic spectra and orthogonality for iterated function system measures, Math. Z. 261 (2009), 373–397.CrossRefGoogle Scholar
10.Fuglede, B., Commuting self-adjoint partial differential operators and a group theoretic problem, J. Funct. Analysis 16 (1974), 101–121.CrossRefGoogle Scholar
11.Hu, T.-Y. and Lau, K.-S., Spectral property of the Bernoulli convolutions, Adv. Math. 219 (2008), 554–567.CrossRefGoogle Scholar
12.Hutchinson, J. E., Fractals and self-similarity, Indiana Univ. Math. J. 30 (1981), 713–747.CrossRefGoogle Scholar
13.Jorgensen, P. E. T., Kornelson, K. and Shuman, K., Orthogonal exponentials for Bernoulli iterated function systems, in Representations, wavelets, and frames, Applied and Numerical Harmonic Analysis, pp. 217–237 (Birkhäuser, Boston, MA, 2008).CrossRefGoogle Scholar
14.Jorgensen, P. E. T. and Pedersen, S., Harmonic analysis of fractal measures, Constr. Approx. 12 (1996), 1–30.CrossRefGoogle Scholar
15.Jorgensen, P. E. T. and Pedersen, S., Dense analytic subspaces in fractal L 2-spaces, J. Analysis Math. 75 (1998), 185–228.Google Scholar
16.Łaba, I. and Wang, Y., On spectral Cantor measures, J. Funct. Analysis 193 (2002), 409–420.CrossRefGoogle Scholar
17.Li, J.-L., Spectral sets and spectral self-affine measures, PhD Thesis, The Chinese University of Hong Kong (2004).Google Scholar
18.Li, J.-L., Spectral self-affine measures in ℝn, Proc. Edinb. Math. Soc. 50 (2007), 197–215.CrossRefGoogle Scholar
19.Li, J.-L., μM,D-orthogonality and compatible pair, J. Funct. Analysis 244 (2007), 628–638.CrossRefGoogle Scholar
20.Li, J.-L., Orthogonal exponentials on the generalized plane Sierpinski gasket, J. Approx. Theory 153 (2008), 161–169.CrossRefGoogle Scholar
21.Li, J.-L., Non-spectral problem for a class of planar self-affine measures, J. Funct. Analysis 255 (2008), 3125–3148.CrossRefGoogle Scholar
22.Li, J.-L., Non-spectrality of planar self-affine measures with three-element digit set, J. Funct. Analysis 257 (2009), 537–552.CrossRefGoogle Scholar
23.Li, J.-L., The cardinality of certain μM, D-orthogonal exponentials, J. Math. Analysis Applic. 362 (2010), 514–522.CrossRefGoogle Scholar
24.Li, J.-L., On the μM, D-orthogonal exponentials, Nonlin. Analysis 73 (2010), 940–951.CrossRefGoogle Scholar
25.Li, J.-L., Spectra of a class of self-affine measures, J. Funct. Analysis 260 (2011), 1086–1095.CrossRefGoogle Scholar
26.Strichartz, R., Remarks on ‘Dense analytic subspaces in fractal L 2-spaces’, J. Analysis Math. 75 (1998), 229–231.Google Scholar
27.Strichartz, R., Mock Fourier series and transforms associated with certain Cantor measures, J. Analysis Math. 81 (2000), 209–238.Google Scholar
28.Strichartz, R., Convergence of mock Fourier series, J. Analysis Math. 99 (2006), 333–353.Google Scholar