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 w 1 u w 2 ⋯ u w n for any sequence w = ( w 1 , w 2 , … , w n ) ∈ { 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 α w ( w ∈ { 1 , 2 , 3 } n ) with 0 < | α w | < 1 , there exists a unique continuous extension f : SG → R of F, such that f ( u w ( x ) ) = α w f ( x ) + h w ( x ) for x ∈ SG , where h w 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 : V n → R .