# Harmonic Structure on Modified Sierpinski Gaskets

Chapter
Part of the Lecture Notes in Electrical Engineering book series (LNEE, volume 126)

## Abstract

How to construct harmonic structures on a class of special self-similar fractals, and then discuss their regularity are important problems in analysis on fractals. J Kigami [3,4,5,6,7] and Strichartz[8,9,10,11] have dicussed in detail. It’s very difficult to built the concept of derivative on fractals directly, therefore we have to consider to construct Laplacians on fractals. The key idea of constructing a Laplacian on fractals is finding a “self-similar” compatible sequence of resistance netwoks on {V m } m ≥ 0. We can start from finite set to built a compatible sequence, then construct a harmonic structure and thus extend to the infinite points. By this way, we could discuss the property of harmonic structures on fractals. In this paper, we study harmonic extension algorithm (matrices) and harmonic structures on modified Sierpinski gaskets (MSG for short). And we also study the relationship between regular harmonic structure and renormalization factor on MSG.

## Keywords

Harmonic Function Sierpinski Gasket Harmonic Structure Harmonic Extension Fractal Interpolation
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## References

1. 1.
Teplyaev, A.: Energy and Laplacian on the Sierpinski Gasket, Fractal Geometry and Applications: A Jubilee of Benoit Mandelbrot, Part 1. In: Proc. Sympos. Pure Math. Amer. Math. Soc., vol. 72, pp. 131–154 (2008)Google Scholar
2. 2.
Ruan, H.-J.: Fractal interpolation functions on p.c.f. self-similar setss. In: Chinese National Mathematics Academic Seminar on Fractal Theory and Dynamic System (2010)Google Scholar
3. 3.
Kigami, J.: A harmonic calculus on the Sierpinski spaces. Japan J.Appl. Math. 8, 259–290 (1989)
4. 4.
Jun, K.: Harmonic calculus on p.c.f. self-similar sets. Trans. Amer. Math. Soc. 335, 721–755 (1993)
5. 5.
Jun, K.: Effective resistances for harmonic structures on p.c.f. self-similar sets. Math. Proc. Cambridge Philos. Soc. 115, 291–303 (1994)
6. 6.
Jun, K.: Analysis on Fractals. Cambridge University Press, London (2001)
7. 7.
Jun, K.: Harmonic Analysis for Resistance Forms. Funct. Anal. 204, 399–444 (2003)
8. 8.
Stanley, J., Strichartz, R., Teplyaev, A.: Energy Partition on Fractals. Indiana University Mathematics Journal 52, 133–156 (2003)
9. 9.
Strichartz, R.S.: The Laplacian on the Sierpinski gasket via the method of averages. Pacific J. Math. 201, 241–256 (2001)
10. 10.
Strichartz, R.S.: Differential equations on fractals: a tutorial. Princeton University Press (2006)Google Scholar
11. 11.
Strichartz, R.: Analysis on fractals. Notices Amer. Math. Soc. 46, 1199–1208 (1999)