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 thatf(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.