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.
Highlights
The Koch curve appeared in a 1904 paper entitled “On a Continuous Curve WithoutTangents, Constructible from Elementary Geometry” by the Swedish mathematician Helge von Koch
The results of [8] and [6] enable us to study Hölder continuity of non-constant harmonic functions on KC and inspire us to ensure that graphs of FIFs generated on KC by non-constant harmonic functions of fractal analysis are attractors of some IFSs
We review some already known results about fractal interpolation on a closed interval in order to derive interpolation functions as attractors of IFSs constructed on KC by non-constant harmonic functions of fractal analysis; see [10], pp. 44–45 or [2], Definition 2.2, p. 44
Summary
Constructible from Elementary Geometry” by the Swedish mathematician Helge von Koch. In [4] the authors showed how one can construct space-filling curves by using hidden variable linear fractal interpolation functions These curves resulted from the projection of the attractor of an iterated function system, or IFS for short. We should use another IFS for the generation of the KC because the proofs of existence and uniqueness of attractor of an IFS constructed on KC are dependent on the first point (or end point) of each curve For this reason, there is an important difference between the proof of uniqueness of invariant set of IFS on KC and the one on a special affine fractal interpolation curve.
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