Inner Products

The discrete inner products of the eigenfunctions are related to each other. Let $e$ and $f$ be eigenfunctions of $-\Delta$ on the Sierpinski Gasket. Pick a positive integer $m$. Then the restrictions of $e$ and $f$ to $V_{m}$ are eigenfunctions of $-\Delta_{m}$ with some eigenvalue $\lambda_{m}$. We found that:

$$<e,f>_{m}=\frac{(6-\lambda_{m})(5-2\lambda_{m})}{(5-\lambda_{m})(2-\lambda_{m})}<e,f>_{m-1}$$

One Consequence of this is that a set eigenfunctions of $-\Delta_{m-1}$ which is orthogonal at level $m-1$ is also orthogonal at level $m$ when extended by spectral decimation.

Since $\frac{2<f,f>_{m}}{3^{m+1}}$ is a Riemann sum approximation for the integral over the Sierpinski Gasket of $f^{2}$, this relation gives us:

$$\frac{2<f,f>_{m}}{3^{m+1}}\prod_{j=m+1}^{\infty}\frac{(6-\lam... ...})}{3(5-\lambda_{j})(2-\lambda_{j})} =\int\limits_{SG}f^{2}d\mu$$

As you can see from the figure:

the correction factor $b(\lambda_{m})=\prod_{j=m+1}^{\infty}\frac{(6-\lambda_{j})(5-2\lambda_{j})}{3(5-\lambda_{j})(2-\lambda_{j})}$ goes to $1$ as $\lambda_m$ goes to $0$. (Calculation of the correction factor assumes that subsequent eigenvalues are calculated with $\epsilon=-1$ unless $\lambda=6$ in which case the first $\epsilon$ equals $1$.)