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.
Polynomials on the Sierpinski Gasket with Respect to Different Laplacians which are Symmetric and Self-Similar
我们研究了Sierpinski垫片（$SG$）上的多项式的模拟（对于某些$n$，解为$Δ^{n+1} u =0$）相对于由方，金，李，和Jehartz构造的一族对称，自相似的拉普拉斯算子，扩展了Needleman，Jehartz，Tejehaev和Yung关于标准Kigami拉普拉斯算子的多项式的工作。我们定义了一个基础的空间多项式，单项式，其特征在于的属性，即一定的“衍生物”是1在一个边界点，而所有其他的“衍生物”消失，我们计算值的单项式在边界点的$SG$。然后，我们提出了一些数据，这表明令人惊讶的关系的单项式的值之间的边界和某些诺依曼特征值的家庭对称自相似拉普拉斯算子。令人惊讶的是，一般情况下的结果与Kigami Laplacian的结果完全不同。
