Fractal Interpolation Using Harmonic Functions on the Koch Curve

Abstract

The Koch curve was first described by the Swedish mathematician Helge von Koch in 1904 as an example of a continuous but nowhere differentiable curve. Such functions are now characterised as fractal since their graphs are in general fractal sets. Furthermore, it can be obtained as the graph of an appropriately chosen iterated function system. On the other hand, a fractal interpolation function can be seen as a special case of an iterated function system thus maintaining all of its characteristics. Fractal interpolation functions are continuous functions that can be used to model continuous signals. An in-depth discussion on the theory of affine fractal interpolation functions generating the Koch Curve by using fractal analysis as well as its recent development including some of the research made by the authors is provided. We ensure that the graph of fractal interpolation functions on the Koch Curve are attractors of an iterated function system constructed by non-constant harmonic functions.

Keywords

• fractal functions
• harmonic functions
• Hölder continuity
• interpolation
• Koch Curve