Higher order tangents and Higher order Laplacians on Sierpinski Gasket Type Fractals

We study higher order tangents and higher order Laplacians on p.c.f. self-similar sets with fully symmetric structures, such as $D3$ or $D4$ symmetric fractals. Firstly, let $x$ be a vertex point in the graphs that approximate the fractal, we prove that for any $f$ defined near $x$, the higher oder weak tangent of $f$ at $x$, if exists, is the uniform limit of local multiharmonic functions that agree with $f$ in some sense near $x$. Secondly, we prove that the higher order Laplacian on a fractal can be expressible as a renormalized uniform limit of higher order graph Laplacians on the graphs that approximate the fractal. The main technical tool is the theory of local multiharmonic functions and local monomials analogous to $(x-x_0)^j/j!$. The results in this paper are closely related to the theory of local Taylor approximations, splines and entire analytic functions. Some of our results can be extended to general p.c.f. fractals. In Appendix of the paper, we provide a recursion algorithm for the exact calculations of the boundary values of the monomials for $D3$ or $D4$ symmetric fractals, which is shorter and more direct than the previous work on the Sierpinski gasket.