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Szegö Limit Theorems on the Sierpiński Gasket

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Abstract

We use the existence of localized eigenfunctions of the Laplacian on the Sierpiński gasket (SG) to formulate and prove analogues of the strong Szegö limit theorem in this fractal setting. Furthermore, we recast some of our results in terms of equally distributed sequences.

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References

  1. Barlow, M.T.: Diffusion on fractals. In: Lectures Notes in Mathematics, vol. 1690. Springer, Berlin (1998)

    Google Scholar 

  2. Barlow, M.T., Kigami, J.: Localized eigenfunctions of the Laplacian on p.c.f. self-similar sets. J. Lond. Math. Soc. 56(2), 320–332 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  3. Dalrymple, K., Strichartz, R.S., Vinson, J.P.: Fractal differential equations on the Sierpinski gasket. J. Fourier Anal. Appl. 5(2/3), 203–284 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  4. Fukushima, M., Shima, T.: On a spectral analysis for the Sierpinski gasket. Potential Anal. 1, 1–35 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  5. Golinskiǐ, B., Ibragimov, I.: A limit theorem of G. Szegö. Math. USSR Izv. 5(2), 421–444 (1971)

    Article  Google Scholar 

  6. Grenander, U., Szegö, G.: Toeplitz Forms and Their Applications, 2nd edn. Chelsea, New York (1984)

    MATH  Google Scholar 

  7. Guillemin, V., Okikiolu, K.: Szegö theorems for Zoll operators. Math. Res. Lett. 3, 1–14 (1996)

    MathSciNet  Google Scholar 

  8. Hirschman, I.I., Jr.: The strong Szegö limit theorem for Toeplitz determinants. Am. J. Math. 88, 577–614 (1966)

    Article  MathSciNet  MATH  Google Scholar 

  9. Kac, M.: Toeplitz matrices, translation kernels and a related problem in probability theory. Duke Math. J. 21, 501–509 (1954)

    Article  MathSciNet  MATH  Google Scholar 

  10. Kigami, J.: Harmonic calculus on p.c.f. self-similar sets. Trans. Am. Math. Soc. 335, 721–755 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  11. Kigami, J.: Analysis on Fractals. Cambridge University Press, New York (2001)

    Book  MATH  Google Scholar 

  12. Laptev, A., Safarov, Y.: Szegö type limit theorems. J. Funct. Anal. 138, 544–559 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  13. Okikiolu, K.: The analogue of the strong Szegö limit theorem on the 2- and 3-dimensional spheres. J. Am. Math. Soc. 9(2), 345–372 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  14. Rammal, R., Toulouse, G.: Random walks on fractal structures and percoloration clusters. J. Phys. Lett. 44, L13–L22 (1983)

    Article  Google Scholar 

  15. Strichartz, R.S.: Analysis on fractals. Not. Am. Math. Soc. 46, 1199–1208 (1999)

    MathSciNet  Google Scholar 

  16. Strichartz, R.S.: Taylor approximations on Sierpinski gasket type fractals. J. Funct. Anal. 174(1), 76–127 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  17. Strichartz, R.S.: Function spaces on fractals. J. Funct. Anal. 198(1), 43–83 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  18. Strichartz, R.S.: Differential Equations on Fractals: A Tutorial. Princeton University Press, Princeton (2006)

    MATH  Google Scholar 

  19. Szegö, G.: On certain Hermitian forms associated with the Fourier series of a positive function. Commun. Sém. Math. Univ. Lund [Medd. Lunds Univ. Mat. Sem.], Tome Supplementaire, pp. 228–238 (1952)

  20. Teplyaev, A.: Spectral analysis on infinite Sierpiński gaskets. J. Funct. Anal. 159, 537–567 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  21. Teplyaev, A.: Gradients on fractals. J. Funct. Anal. 174(1), 128–154 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  22. Widom, H.: Szegö’s theorem and a complete symbolic calculus for pseudo-differential operators. In: Seminar on Singularities of Solutions. Princeton University Press, Princeton (1979), pp. 261–283

    Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Mathematics, University of Maryland, College Park, MD, 20742-4015, USA

    Kasso A. Okoudjou

  2. Department of Mathematics, University of Connecticut, Storrs, CT, 06269-3009, USA

    Luke G. Rogers

  3. Department of Mathematics, Malott Hall, Cornell University, Ithaca, NY, 14853-4201, USA

    Robert S. Strichartz

Authors
  1. Kasso A. Okoudjou
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  2. Luke G. Rogers
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  3. Robert S. Strichartz
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Corresponding author

Correspondence to Kasso A. Okoudjou.

Additional information

Communicated by Hans Feichtinger.

Research of the third author supported in part by the National Science Foundation, grant DMS-065440.

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Okoudjou, K.A., Rogers, L.G. & Strichartz, R.S. Szegö Limit Theorems on the Sierpiński Gasket. J Fourier Anal Appl 16, 434–447 (2010). https://doi.org/10.1007/s00041-009-9102-0

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  • DOI: https://doi.org/10.1007/s00041-009-9102-0

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