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Originally, we calculated the functions as they were defined in the first section. However, we eventually decided on an easier method.
Suppose is an orthonormal basis of eigenfunctions, with respect to the inner product
(the discrete inner product on the th level of the gasket).
, then observe:
since 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.