Fractal interpolation on the Sierpinski Gasket

ÇELİK D., Kocak S., ÖZDEMİR Y.

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS, vol.337, no.1, pp.343-347, 2008 (SCI-Expanded) identifier identifier

  • Publication Type: Article / Article
  • Volume: 337 Issue: 1
  • Publication Date: 2008
  • Doi Number: 10.1016/j.jmaa.2007.03.102
  • Journal Name: JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus
  • Page Numbers: pp.343-347
  • Keywords: fractal interpolation, Sierpinski Gasket, harmonic functions, self-similar functions
  • Anadolu University Affiliated: Yes

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