Polynomial Type Functions and Power Series on the
Sierpinski Gasket
Some basic definitons need to be defined first in order to understand polynomials and power series on the Sierpinski Gasket(SG). Let u be a function defined on SG. Then the pointwise laplacian is defined as:

$\displaystyle \Delta u(x)=\lim_{m \to \infty}\left(\frac{3}{2}\right)5^m\sum_{x \sim_m y}(u(y)-u(x))$


Where $ x \sim_m y$ means that x is connected to y on the level m gasket approximation.

The normal derivative at the boundary $ x \in V_0$ is:

$\displaystyle \partial_n u(x)=\lim_{m \to \infty}\left(\frac{5}{3}\right)^m \sum_{x \sim_m y}(u(y)-u(x))$



And the tangential derivative at v1 is:

$\displaystyle \partial_T u(v_1)=\lim_{m \to \infty}5^m(u(F_2^m v_2)-u(F_3^m v_3))$

Where $ F_i$ is a contraction toward the boundary point $ v_i$ by $ \frac{1}{2}$.

There is also a form of the Gauss-Green formula for functions in the domain of the laplacian.

$\displaystyle \int (u \Delta v -\Delta u v)=\sum_{x \in V_0}\left(u(x)\partial_n v(x)-\partial_n u(x) v(x)\right)$

Where $ V_0$ is the boundary of SG.

To define polynomials on SG we will use the idea from the real numbers that after $ n+1$ derivatives opperate on a $ n$ degree polynomial, the result is the constant 0 function. For SG we will define polynomials, or n-harmonic funtions as those functions u so that $ \Delta^n u =0$. There is an easy basis for these functions:

$\displaystyle \Delta^n f_{jk}(v_i)=\delta_{nj}\delta_{ki}\notag$
for$\displaystyle \quad 1 \leq i,k \leq 3$ and$\displaystyle \quad j\geq0$


These $ 3n$ boundary conditions uniquely detemine each function $ f_{jk}$. However, these boundary condtions are defined for three different boundary points. When looking at power series it is important to expand about a single point. To do this we defined a polynomial basis for n-harmonic functions.

The polynomial basis defined around the boundary point $ v_i$ is:

$\displaystyle \Delta^m P_{j1}^i(v_i)$ $\displaystyle =\delta_{jm}$ $\displaystyle \Delta^m P_{j2}^{(i)}(v_i)$ $\displaystyle =0$ $\displaystyle \Delta^m P_{j3}^i(v_i)$ $\displaystyle =0$
$\displaystyle \partial_n \Delta^m P_{j1}^i(v_i)$ $\displaystyle =0$ $\displaystyle \partial_n\Delta^m P_{j2}^i(v_i)$ $\displaystyle =\delta_{jm}$ $\displaystyle \partial_n\Delta^m P_{j3}^i(v_i)$ $\displaystyle =0\notag$
$\displaystyle \partial_T\Delta^m P_{j1}^i(v_i)$ $\displaystyle =0$ $\displaystyle \partial_T\Delta^m P_{j2}^i(v_i)$ $\displaystyle =0$ $\displaystyle \partial_T\Delta^m P_{j3}^i(v_i)$ $\displaystyle =\delta_{jm}\notag$


To take a look at what these functions look like click below.
P0_1 P0_2 P0_3
P1_1 P1_2 P1_3
P2_1 P2_2 P2_3
P3_1 P3_2 P3_3
P4_1 P4_2 P4_3

The Pj2 pictures are misleading. At first they appear to be all 1 sign. But after a while they tend into a form that looks more like this:
P12_2


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