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Dirichlet Kernels

We also generated the Dirichlet kernels on the gasket. They differ from the $\psi$ functions in that the orthonormal basis is done with respect to the inner product on the full gasket: $<f,g>_SG=\int\limits_{SG}fg d\mu$, not the discrete inner product on a level of the gasket. So $D^{m}_{y}(x)=\sum_{j=1}^{N_m} \frac{3^{m+1}}{2b_m(\lambda)} u_j(x)u_j(y)$. Here is an example Dirichlet kernel:

One might note that it looks like the $\psi$ functions. This is related to the fact that the $\psi$ functions fall off exponentially.

Brain Street 2001-11-11