Generalized eigenfunctions and a Borel Theorem on the Sierpinski Gasket.∗

There is a well developed theory (see [5, 9]) of analysis on certain types of fractal sets, of which the Sierpinski Gasket (SG) is the simplest non-trivial example. In this theory the fractals are viewed as limits of graphs, and notions analogous to the Dirichlet energy and the Laplacian are constructed as renormalized limits of the corresponding objects on the approximating graphs. The nature of this construction has naturally led to extensive study of the eigenfunctions of this Laplacian, and to functional-analytic notions based on the eigenfunctions. However, more recent work [7, 2] has examined other elementary functions on SG, including analogues of polynomials, analytic functions and certain exponentials. A forthcoming paper [8] will extend this investigation to study smooth bump functions and a method for partitioning smooth functions subordinate to an open cover. In the present work we prove there are exponentially decaying generalized eigenfunctions on a blow-up of S G with boundary (which we denote S G∞), proving: Theorem 1.1. For each λ < 0 and j ∈ N there is a smooth function E j λ on S G∞ such that for each j we have (∆ + λ)E j λ = − jE j−1 λ . Moreover E j λ decays exponentially away from the boundary point of S G∞ and satisfies |E j λ| ≤ j!|λ|− j.