Abstract
Surprisingly, Fourier series on certain fractals can have better convergence properties than classical Fourier series. This is a result of the existence of gaps in the spectrum of the Laplacian. In this work we prove general criteria for the existence of gaps when the Laplacian admits spectral decimation. The known examples, including the Sierpinski gasket and the level-3 Sierpinski gasket, and the new examples including the fractal-3 tree, the Hexagasket and the infinite family of tree-like fractals satisfy the criteria.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Adams, B., Smith, S.A., Strichartz, R., Teplyaev, A.: The Spectrum of the Laplacian on the Pentagasket. Trends in Mathematics, Fractals in Graz 2001. Birkhauser, Basel (2003), pp. 1–24
Bajorin, N., Chen, T., Dagan, A., Emmons, C., Hussein, M., Khalil, M., Mody, P., Steinhurst, B., Teplyaev, A.: Vibration modes of 3n-gaskets and other fractals. J. Phys. A: Math. Theor. 41 (2008)
Barlow, M.: Diffusion on Fractals. Lecture Notes Math., vol. 1690. Springer, Berlin (1998)
Barlow, M., Perkins, E.: Brownian motion on the Sierpinski gasket. Probab. Theory Relat. Fields 79, 543–623 (1988)
Drenning, S.: Fractal analysis: Extending the domain. Cornell Univ. Math. Dept. Senior Thesis (2005)
Drenning, S., Strichartz, R.: Spectral decimation on Hambly’s homogeneous hierarchical gaskets. Preprint
Duong, X.T., Ouhabaz, E.M., Sikora, A.: Plancherel type estimates and sharp spectral multipliers. J. Funct. Anal. Appl. 196, 443–485 (2002)
Fukushima, M., Shima, T.: On a spectral analysis for the Sierpinski gasket. Potential Anal. 1, 1–35 (1992)
Gibbons, M., Raj, A., Strichartz, R.: The finite element method on the Sierpinski gasket. Constr. Approx. 17, 561–588 (2001)
Hambly, B., Kumagai, T.: Transition density estimates for diffusion processes on post critically finite self-similar fractals. Proc. Lond. Math. Soc. 78, 431–458 (1999)
Kigami, J.: Harmonic calculus on p.c.f. self-similar sets. Trans. Am. Math. Soc. 335, 721–755 (1993)
Kigami, J.: Analysis on Fractals. Cambridge Univ. Press, New York (2001)
Kigami, J., Lapidus, M.L.: Weyl’s problem for the spectral distribution of Laplacians on p.c.f. self-similar fractals. Commun. Math. Phys. 158, 93–125 (1993)
Malozemov, L., Teplyaev, A.: Self-similarity, operators and dynamics. Math. Phys. Anal. Geom. 6, 201–218 (2003)
Milnor, J.: Dynamics in one Complex Variable, 3rd edn. Princeton Univ. Press, New Jersey (2006)
Shima, T.: On eigenvalue problems for Laplacians on p.c.f. self-similar sets. Jpn. J. Ind. Appl. Math. 13, 1–23 (1996)
Strichartz, R.: Laplacians on fractals with spectral gaps have nicer Fourier series. Math. Res. Lett. 12, 269–274 (2005)
Strichartz, R.: Differential Equations on Fractals: A Tutorial. Princeton Univ. Press, New Jersey (2006)
Teplyaev, A.: Spectral analysis on infinite Sierpinski gasket. J. Funct. Anal. 159, 537–567 (1998)
Zhou, D.: Spectral analysis of Laplacians on the Vicsek set. Pac. J. Math. 241(2), 369–398 (2009)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Robert Strichartz.
Rights and permissions
About this article
Cite this article
Zhou, D. Criteria for Spectral Gaps of Laplacians on Fractals. J Fourier Anal Appl 16, 76–96 (2010). https://doi.org/10.1007/s00041-009-9087-8
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00041-009-9087-8