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# What is sampling? (aka What are these functions I keep hearing about?)

Sampling is the practice of reconstructing a function from its values at a finite number of points. The simplest case is to consider is a function . Clearly, not every function can be reconstructed, as seen in the example below:

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.

Next: Computation of the functions Up: sampling Previous: sampling
Brain Street 2001-11-11