Hostname: page-component-cd9895bd7-jkksz Total loading time: 0 Render date: 2024-12-26T15:10:29.488Z Has data issue: false hasContentIssue false
  • English
  • Français

Generalized Eigenfunctions and a Borel Theorem on the Sierpinski Gasket

Published online by Cambridge University Press:  20 November 2018

Kasso A. Okoudjou ,
Luke G. Rogers  and
Robert S. Strichartz
Kasso A. Okoudjou
Affiliation:
Mathematics Department, Cornell University, Ithaca, NY 14853, U.S.A. e-mail: kasso@math.umd.edurogers@math.uconn.edustr@math.cornell.edu
Luke G. Rogers
Affiliation:
Mathematics Department, Cornell University, Ithaca, NY 14853, U.S.A. e-mail: kasso@math.umd.edurogers@math.uconn.edustr@math.cornell.edu
Robert S. Strichartz
Affiliation:
Mathematics Department, Cornell University, Ithaca, NY 14853, U.S.A. e-mail: kasso@math.umd.edurogers@math.uconn.edustr@math.cornell.edu
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.

We prove there exist exponentially decaying generalized eigenfunctions on a blow-up of the Sierpinski gasket with boundary. These are used to show a Borel-type theorem, specifically that for a prescribed jet at the boundary point there is a smooth function having that jet.

Keywords

28A8031C45
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 52 , Issue 1 , 01 March 2009 , pp. 105 - 116
Copyright
Copyright © Canadian Mathematical Society 2009

References

[1] Barlow, M. T. and Kigami, J., Localized eigenfunctions of the Laplacian on p.c.f. self-similar sets. J. London Math. Soc. (2) 56(1997), no. 2, 320–332.Google Scholar
[2] Ben-Gal, N., Shaw-Krauss, A., Strichartz, R. S., and Young, C., Calculus on the Sierpinski gasket II. Point singularities, eigenfunctions, and normal derivatives of the heat kernel. Trans. Amer. Math. Soc. 358(2006), no. 9, 3883–3936 (electronic).Google Scholar
[3] Dalrymple, K., Strichartz, R. S., and Vinson, J. P., Fractal differential equations on the Sierpinski gasket, J. Fourier Anal. Appl. 5(1999), no. 2-3, 203–284.Google Scholar
[4] Fukushima, M. and Shima, T., On a spectral analysis for the Sierpiński gasket. Potential Anal. 1(1992), no. 1, 1–35.Google Scholar
[5] Kigami, J., Analysis on fractals. Cambridge Tracts in Mathematics 143, Cambridge University Press, Cambridge, 2001.Google Scholar
[6] Malozemov, L. and Teplyaev, A., Self-similarity, operators and dynamics. Math. Phys. Anal. Geom. 6(2003), no. 3, 201–218.Google Scholar
[7] Needleman, J., Strichartz, R. S., Teplyaev, A., and Yung, P.-L., Calculus on the Sierpinski gasket. I. Polynomials, exponentials and power series. J. Funct. Anal. 215(2004), no. 2, 290–340.Google Scholar
[8] Rogers, L. G., Strichartz, R. S., and Teplyaev, A., Smooth bumps, a Borel theorem and partitions of unity on p.c.f. fractals, To appear, Trans. Amer.Math. Soc.Google Scholar
[9] Strichartz, R. S., Differential equations on fractals: A tutorial. Princeton University Press, Princeton, NJ, 2006.Google Scholar
[10] Strichartz, R. S., Fractals in the large. Canad. J. Math. 50(1998), no. 3, 638–657.Google Scholar
[11] Teplyaev, A., Spectral analysis on infinite Sierpiński gaskets. J. Funct. Anal. 159(1998), no. 2, 537–567.Google Scholar