Eigenfunctions of the Laplacian
Consider the eigenfunctions on the real line of the Laplacian. These are the solutions to the equation f''+af=0, which are always either exponentials or trigonometric functions. They are of tremendous interest in both science and math, and together with the multiharmonic functions on the real line (the polynomials), they form all the familiar functions and are the starting points for many more bizarre functions.
With this in mind, eigenfunctions of the Laplacian on the infinite Sierpinski gasket are also of major interest. Given the eigenvalue a<0 (the cases of a=2,4,5 will be of interest later), the space of eigenfunctions with eigenvalue a is 3 dimensional. For the case of the infinite blowup in one direction mentioned earlier, there is a very "nice" basis for this space, Ca, Sa, and Qa, with the following useful properties:
Ca(xn)=1-an/4
Sa(xn)=-an(Productk>or=0(1+4/(2-an-k))/4
Qa(xn)=-.75an/a
limn-> -infinitySa(xn)/Ca(xn)=1
And Ca,Sa are even while Qa is odd.
In the formulas above [...,a-1, a0, a1,...] is the doubly infinite sequence of eigenvalues of graph Laplacians, which extends in two directions now because the gasket both contracts inward and zooms out. Remember how an-1=an(5-an). Also, note that since a<0, the sequence is uniquely determined by a0 since the ai can no longer bifurcate going in..
The fact that the blowup these functions lie on has a boundary point is what was used to determine this basis. The derivation is in the "Calculus..." paper, which involves power series.However, you can see using the formulas that the graph eigenvalue equations are fufilled by graphs with the proper values at the vertices.
Using this special basis, a decaying exponential function and a level function can be constructed:
Ea=Ca-Sa
La=Ca-aQa/3
Ea decays in all directions since it is even and decaying on the horizntal line. La has the value 1 on all xn. It can be shown easily that La has the value 1 on the entire bottom horizontal, and is periodic on any horizontal.