Using an orthonormal basis

Originally, we calculated the $\psi$ functions as they were defined in the first section. However, we eventually decided on an easier method. Suppose $\{u_j\}$ is an orthonormal basis of eigenfunctions, with respect to the inner product $<f,g>=\sum_{y\in V_m-V_0} f(y)g(y)$ (the discrete inner product on the $m$th level of the gasket). Suppose $\psi_y(x)=\sum_j u_j(y) u_j(x)$, then observe:

$$\sum_{x\in V_m-V_0} \psi_y(x)f(x) = \sum_j u_j(y)\sum_x u_j(x)f(x) = \sum_j <f,u_j>u_j(y) = f(y)$$

since $u_j$ is an orthonormal basis.
So all that remains to be done, is to find an orthonormal basis for our space of eigen functions. But, since eigenfunctions with different eigenvalues are already orthogonal, all we must do is apply the Gram method to the different eigenspaces. You may see pictures and data of the orthonormal basis here.