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Analysis of the Laplacian and spectral operators on the Vicsek set
1. | Department of Mathematics, Princeton University, Fine Hall, Washington Road, Princeton NJ 08544-1000, USA Government |
2. | Mathematics Department, Malott Hall, Cornell Univeristy, Ithaca, NY 14853, United States |
3. | Department of Mathematics, Brown University, Box 1917, Providence, RI 02912, USA Government |
References:
[1] |
Bryant Adams, S. Alex Smith, Robert S. Strichartz and Alexander Teplyaev, The spectrum of the Laplacian on the pentagasket, Fractals in Graz 2001, Trends Math., Birkhäuser, Basel, 2003, pp. 1-24. |
[2] |
Adam Allan, Michael Barany and Robert S. Strichartz, Spectral operators on the Sierpinski gasket I, Complex variables and elliptic operators, 54 (2009), 521-543. |
[3] |
Tyrus Berry, Steven Heilman and Robert S. Strichartz, Outer approximation of the spectrum of a fractal Laplacian, Experimental Mathematics, 18 (2009), 449-480, arXiv:0904.3757. |
[4] |
Brian Bockelman and Robert S. Strichartz, Partial differential equations on products of Sierpinski gaskets, Indiana Univ. Math. J., 56 (2007), 1361-1375.
doi: doi:10.1512/iumj.2007.56.2981. |
[5] |
Kevin Coletta, Kealey Dias and Robert S. Strichartz, Numerical analysis on the Sierpinski gasket, with applications to Schrödinger equations, wave equation, and Gibbs' phenomenon, Fractals, 12 (2004), 413-449.
doi: doi:10.1142/S0218348X04002689. |
[6] |
Sarah Constantin, Robert S. Strichartz and Wheeler Miles, Spectral operators on vicsek sets, 2009, http://www.math.cornell.edu/ mhw33. |
[7] |
Kyallee Dalrymple, Robert S. Strichartz and Jade P. Vinson, Fractal differential equations on the Sierpinski gasket, J. Fourier Anal. Appl., 5 (1999), 203-284.
doi: doi:10.1007/BF01261610. |
[8] |
S. Drenning and Robert S. Strichartz, Spectral decimation on Hambly's homogeneous, hierarchical gaskets, Ill. J. Math., 53 (2009), 915-937. |
[9] |
Xuan Thinh Duong, El Maati Ouhabaz and Adam Sikora, Plancherel-type estimates and sharp spectral multipliers, J. Funct. Anal., 196 (2002), 443-485.
doi: doi:10.1016/S0022-1236(02)00009-5. |
[10] |
Taryn Flock and Robsert S. Strichartz, Laplacians on a family of quadratic Julia sets, Trans. Amer. Math. Soc., to appear. |
[11] |
M. Fukushima and T. Shima, On a spectral analysis for the Sierpiński gasket, Potential Anal., 1 (1992), 1-35.
doi: doi:10.1007/BF00249784. |
[12] |
Peter J. Grabner and Wolfgang Woess, Functional iterations and periodic oscillations for simple random walk on the Sierpiński graph, Stochastic Process. Appl., 69 (1997), 127-138.
doi: doi:10.1016/S0304-4149(97)00033-1. |
[13] |
A. Grigor'yan and L. Saloff-Coste, Heat kernels on manifolds with ends, Ann. Inst. Fourier, 59 (2009), 1917-1997. |
[14] |
Kathryn E. Hare and Denglin Zhou, Gaps in the ratios of the spectrum of Laplacians on fractals, Fractals, 17 (2009), 523-535.
doi: doi:10.1142/S0218348X0900451X. |
[15] |
Jun Kigami, "Analysis on Fractals," Cambridge Tracts in Mathematics, vol. 143, Cambridge University Press, Cambridge, 2001. |
[16] |
Jun Kigami and Michel L. Lapidus, Weyl's problem for the spectral distribution of Laplacians on p.c.f. self-similar fractals, Comm. Math. Phys., 158 (1993), 93-125.
doi: doi:10.1007/BF02097233. |
[17] |
Richard Oberlin, Brian Street and Robert S. Strichartz, Sampling on the Sierpinski gasket, Experiment. Math., 12 (2003), 403-418. |
[18] |
Adam Sikora, Multivariable spectral multipliers and analysis of quasielliptic operators on fractals, Indiana Univ. Math. J., 58 (2009), 317-334.
doi: doi:10.1512/iumj.2009.58.3745. |
[19] |
Elias M. Stein, "Singular Integrals and Differentiability Properties of Functions," Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. |
[20] |
Robert S. Strichartz, Laplacians on fractals with spectral gaps have nicer Fourier series, Math. Res. Lett., 12 (2005), 269-274. |
[21] |
Robert S. Strichartz, "Differential Equations on Fractals: A Tutorial," Princeton University Press, Princeton, NJ, 2006. |
[22] |
Alexander Teplyaev, Spectral analysis on infinite Sierpiński gaskets, J. Funct. Anal., 159 (1998), 537-567.
doi: doi:10.1006/jfan.1998.3297. |
[23] |
Denglin Zhou, Spectral analysis of Laplacians on the Vicsek set, Pacific J. Math., 241 (2009), 369-398.
doi: doi:10.2140/pjm.2009.241.369. |
show all references
References:
[1] |
Bryant Adams, S. Alex Smith, Robert S. Strichartz and Alexander Teplyaev, The spectrum of the Laplacian on the pentagasket, Fractals in Graz 2001, Trends Math., Birkhäuser, Basel, 2003, pp. 1-24. |
[2] |
Adam Allan, Michael Barany and Robert S. Strichartz, Spectral operators on the Sierpinski gasket I, Complex variables and elliptic operators, 54 (2009), 521-543. |
[3] |
Tyrus Berry, Steven Heilman and Robert S. Strichartz, Outer approximation of the spectrum of a fractal Laplacian, Experimental Mathematics, 18 (2009), 449-480, arXiv:0904.3757. |
[4] |
Brian Bockelman and Robert S. Strichartz, Partial differential equations on products of Sierpinski gaskets, Indiana Univ. Math. J., 56 (2007), 1361-1375.
doi: doi:10.1512/iumj.2007.56.2981. |
[5] |
Kevin Coletta, Kealey Dias and Robert S. Strichartz, Numerical analysis on the Sierpinski gasket, with applications to Schrödinger equations, wave equation, and Gibbs' phenomenon, Fractals, 12 (2004), 413-449.
doi: doi:10.1142/S0218348X04002689. |
[6] |
Sarah Constantin, Robert S. Strichartz and Wheeler Miles, Spectral operators on vicsek sets, 2009, http://www.math.cornell.edu/ mhw33. |
[7] |
Kyallee Dalrymple, Robert S. Strichartz and Jade P. Vinson, Fractal differential equations on the Sierpinski gasket, J. Fourier Anal. Appl., 5 (1999), 203-284.
doi: doi:10.1007/BF01261610. |
[8] |
S. Drenning and Robert S. Strichartz, Spectral decimation on Hambly's homogeneous, hierarchical gaskets, Ill. J. Math., 53 (2009), 915-937. |
[9] |
Xuan Thinh Duong, El Maati Ouhabaz and Adam Sikora, Plancherel-type estimates and sharp spectral multipliers, J. Funct. Anal., 196 (2002), 443-485.
doi: doi:10.1016/S0022-1236(02)00009-5. |
[10] |
Taryn Flock and Robsert S. Strichartz, Laplacians on a family of quadratic Julia sets, Trans. Amer. Math. Soc., to appear. |
[11] |
M. Fukushima and T. Shima, On a spectral analysis for the Sierpiński gasket, Potential Anal., 1 (1992), 1-35.
doi: doi:10.1007/BF00249784. |
[12] |
Peter J. Grabner and Wolfgang Woess, Functional iterations and periodic oscillations for simple random walk on the Sierpiński graph, Stochastic Process. Appl., 69 (1997), 127-138.
doi: doi:10.1016/S0304-4149(97)00033-1. |
[13] |
A. Grigor'yan and L. Saloff-Coste, Heat kernels on manifolds with ends, Ann. Inst. Fourier, 59 (2009), 1917-1997. |
[14] |
Kathryn E. Hare and Denglin Zhou, Gaps in the ratios of the spectrum of Laplacians on fractals, Fractals, 17 (2009), 523-535.
doi: doi:10.1142/S0218348X0900451X. |
[15] |
Jun Kigami, "Analysis on Fractals," Cambridge Tracts in Mathematics, vol. 143, Cambridge University Press, Cambridge, 2001. |
[16] |
Jun Kigami and Michel L. Lapidus, Weyl's problem for the spectral distribution of Laplacians on p.c.f. self-similar fractals, Comm. Math. Phys., 158 (1993), 93-125.
doi: doi:10.1007/BF02097233. |
[17] |
Richard Oberlin, Brian Street and Robert S. Strichartz, Sampling on the Sierpinski gasket, Experiment. Math., 12 (2003), 403-418. |
[18] |
Adam Sikora, Multivariable spectral multipliers and analysis of quasielliptic operators on fractals, Indiana Univ. Math. J., 58 (2009), 317-334.
doi: doi:10.1512/iumj.2009.58.3745. |
[19] |
Elias M. Stein, "Singular Integrals and Differentiability Properties of Functions," Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. |
[20] |
Robert S. Strichartz, Laplacians on fractals with spectral gaps have nicer Fourier series, Math. Res. Lett., 12 (2005), 269-274. |
[21] |
Robert S. Strichartz, "Differential Equations on Fractals: A Tutorial," Princeton University Press, Princeton, NJ, 2006. |
[22] |
Alexander Teplyaev, Spectral analysis on infinite Sierpiński gaskets, J. Funct. Anal., 159 (1998), 537-567.
doi: doi:10.1006/jfan.1998.3297. |
[23] |
Denglin Zhou, Spectral analysis of Laplacians on the Vicsek set, Pacific J. Math., 241 (2009), 369-398.
doi: doi:10.2140/pjm.2009.241.369. |
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