One will note that at the marked points, both the red and the blue functions are , but only one is the constant function , hence sampling would fail at these points.

Hence, some restrictions must be put on . For a sufficiently nice function on , with , may be
expanded into a sine series:

.

One says is band-limited with bandwidth if . In this case, one can see that in fact,

Hence, we see that bandlimited functions of bandwidth , can be reconstructed by sampling from points.

We wish to study the analogous concept on the Sierpinski Gasket. First, one must note that , and hence is an eigenfunction of the laplacian of eigenvalue . Moreover, , and so we say that has Dirichlet boundary conditions. Hence, for a function defined on the Sierpinski Gasket, we say is bandlimited if , where and .

It makes the most sense to sample on , the th level of the Sierpinski Gasket. One wishes to find functions such that, for a band limited function , .

Hence, we wish to have , and for . Consider the matrix , letting and vary. This matrix is square and invertible due to a result by Shima and Fukashima. Noting that , we see that we ought to define . Since this was essentially a change of basis, clearly we will be able to reconstruct bandlimited functions with these functions.