Abstract
In this article, firstly, an overview of affine fractal interpolation functions using a suitable iterated function system is presented and, secondly, the construction of Bernstein affine fractal interpolation functions in two and three dimensions is introduced. Moreover, the convergence of the proposed Bernstein affine fractal interpolation functions towards the data generating function does not require any condition on the scaling factors. Consequently, the proposed Bernstein affine fractal interpolation functions possess irregularity at any stage of convergence towards the data generating function.
Highlights
Classic interpolation techniques fit an elementary function to the given data in order to render a connected visualisation of a sample
Axioms 2020, 9, 119 interpolation theory and classical Bernstein polynomial, we construct a sequence of Bernstein affine fractal interpolation functions in one and two variables that uniformly converges to the data generating function for any choice of the scaling factors
If the magnitude of the scaling factors goes to zero, the corresponding existing affine FIFs converge to the data generating function
Summary
Classic interpolation techniques fit an elementary function to the given data in order to render a connected visualisation of a sample. Utilising the theory of an iterated function system firstly presented in [1] and popularised in [2,3] the concept of a fractal interpolation function was proposed whose graph is the attractor, a fractal set, of an appropriately chosen iterated function system If this graph has a Hausdorff–Besicovitch dimension between 1 and 2, the resulting attractor is called fractal interpolation curved line or fractal interpolation curve. Axioms 2020, 9, 119 interpolation theory and classical Bernstein polynomial, we construct a sequence of Bernstein affine fractal interpolation functions in one and two variables that uniformly converges to the data generating function for any choice of the scaling factors.
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