Irregular Sampling

Suppose we don't know the values of a bandlimited function on $V_m-V_0$, but instead on $X\subset \cup_{i=1}^{\infty} V_i$, $\vert X\vert=\vert V_m-V_0\vert$. When can we reconstruct the function? If one looks back to how the $\psi$ functions were defined in the first place, one would realize that band limited functions could be reconstructed, whenever $A=(\phi_k\vert _X(y))$ is invertible. On the interval $[0,\pi]$, this is always true. Not so on the gasket:

Consider the following eigenfunction of the discrete inner product on the first level of the gasket:

This is an eigenfunction of eigenvalue 5. This means it corresponds to an eigenfunction of the laplacian on the gasket, and hence is a bandlimited function. But one will notice, that by symmetry, this eigenfunction is 0 all along the center.

So if one were to sample just at points down the center (no matter how many sampling points were used) this function would look just like the zero function.

Hence, this function cannot be reconstructed if the sampling points are just down the center.

We conjecture that if the sampling points are sufficiently well distributed, $det(A)$ will stay bounded from $0$. The data showing determinants can be found here.

What do we mean by sufficiently well distributed? Suppose we want to move around the three red dots in the picture below. Consider the bottom dot. It would be allowed to range over the blue area, and the determinant would stay bounded from zero. If the dot were allowed to vary into the green area (as the other dots were going similar things) but not onto the pink points, then the determinant would stay non-zero, but get arbitrarily close to zero. This concept is easily extended to lower levels of the gasket.