Fractal interpolation on the Sierpinski Gasket

Abstract

We prove for the Sierpinski Gasket (SG) an analogue of the fractal interpolation theorem of Barnsley. Let V-0 = [p(1), p(2), p(3)) be the set of vertices of SG and u(i) (x) = 1/2 (x + p(i)) the three contractions of the plane, of which the SG is the attractor. Fix a number n and consider the iterations u(w) = u(w1) u(w2)...u(wn) for any sequence w = (w(1), w(2),...,w(n),) is an element of{1, 2, 3}(n). The union of the images of V-0 under these iterations is the set of nth stage vertices V-n of SG. Let F : V-n -> R be any function. Given any numbers ce", (W is an element of {1, 2, 3}(n)) with 0 < vertical bar alpha(w)vertical bar < 1, there exists a unique continuous extension f : SG -> R of F, such that f(u(w)(x)) = alpha(w) f(x) + h(w)(x) for X is an element of SG, where h(w) are harmonic functions on SG for w is an element of {1, 2, 3}(n). Interpreting the harmonic functions as the "degree I polynomials" on SG is thus a self-similar interpolation obtained for any start function F:V-n -> R (C) Elsevier Inc. All rights reserved.