Taylor Approximations on Sierpinski Gasket Type Fractals

Abstract

For a class of fractals that includes the familiar Sierpinski gasket, there is now a theory involving Laplacians, Dirichlet forms, normal derivatives, Green's functions, and the Gauss–Green integration formula, analogous to the theory of analysis on manifolds. This theory was originally developed as a by-product of the construction of stochastic processes analogous to Brownian motion, but has been given by a direct analytic construction in the work of Kigami. Until now, this theory has not provided anything analogous to the gradient of a function, or a local Taylor approximation. In this paper we construct a family of derivatives, which includes the known normal derivative, at vertex points in the graphs that approximate the fractal, and obtain Taylor approximations at these points. We show that a function in the domain of Δn can be locally well approximated by an n-harmonic function (solution of Δnu=0). One novel feature of this result is that it requires several different estimates to describe the optimal rate of approximation.