| A generalized theory of harmonic analysis on fractals is
constructed for a class of self-similar fractals. This theory is then applied to the Sierpinski Gasket.
A graph approximating the Sierpinski gasket is defined by an iterated function system. The closure of
this sequence of graphs is the Sierpinski Gasket. Notions analogous to energy, harmonic functions, and
the Laplacian are described as the limit of discrete forms on the graphs. From these, three
formulations of the second derivative are examined. Comparisons are made to the unit interval. The
eigenvalues of the harmonic functions can be similarly computed from the limit of graphs, and the
spectrum is examined. Time permitting, current directions of research will be discussed. |