Abstract
We introduce the novel concept of a non-stationary iterated function system by considering a countable sequence of distinct set-valued maps { F k } k ∈ N where each F k maps H ( X ) → H ( X ) and arises from an iterated function system. Employing the recently-developed theory of non-stationary versions of fixed points and the concept of forward and backward trajectories, we present new classes of fractal functions exhibiting different local and global behavior and extend fractal interpolation to this new, more flexible setting.
Highlights
Contractive operators on complete function spaces play a crucial role in the theory of differential and integral equations and are essential for the development of iterative solvers
One class of such contractive operators is defined on the graphs of functions using a particular type of iterated function system (IFS)
There exists an extensive literature on IFSs and fractal functions including, for instance, [1,2,3]
Summary
Contractive operators on complete function spaces play a crucial role in the theory of differential and integral equations and are essential for the development of iterative solvers One class of such contractive operators is defined on the graphs of functions using a particular type of iterated function system (IFS). Motivated by non-stationary subdivision algorithms, a more general class of sequences consisting of different contractive operators was introduced in [4] and their limit properties studied. These ideas were extended in [5] to sequences of different contractive operators mapping between different spaces. This article uses the aforementioned new ideas to introduce the novel concept of non-stationary IFS and non-stationary fractal interpolation.
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