Fractal interpolation on the Sierpinski Gasket

We prove for the Sierpinski Gasket (SG) an analogue of the fractal interpolation theorem of Barnsley. Let V0 = {p1,p2,p3} be the set of vertices of SG and ui(x) = 12 (x + pi ) the three contractions of the plane, of which the SG is the attractor. Fix a number n and consider the iterations uw = uw1uw2 ···uwn for any sequence w = (w1,w2, . . . , wn) ∈ {1, 2, 3}n. The union of the images of V0 under these iterations is the set of nth stage vertices Vn of SG. Let F : Vn →R be any function. Given any numbers αw (w ∈ {1, 2, 3}n) with 0 < |αw| < 1, there exists a unique continuous extension f :SG→R of F, such that f uw(x) = αwf (x)+ hw(x) for x ∈ SG, where hw are harmonic functions on SG for w ∈ {1, 2, 3}n. Interpreting the harmonic functions as the “degree 1 polynomials” on SG is thus a self-similar interpolation obtained for any start function F :Vn→R. © 2007 Elsevier Inc. All rights reserved.