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The discrete inner products of the eigenfunctions are related to each
other. Let and be eigenfunctions of on the Sierpinski Gasket.
Pick a positive integer . Then the restrictions of and to are
eigenfunctions of with some eigenvalue . We found that:
One Consequence of this is that a set eigenfunctions of which is orthogonal at
level is also orthogonal at level when extended by spectral decimation.
is a Riemann sum approximation for the integral over the
Sierpinski Gasket of , this relation gives us:
As you can see from the figure:
the correction factor
goes to as goes to . (Calculation of the correction factor assumes that
subsequent eigenvalues are calculated with unless in which case the first equals .)