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Discontinuity, Nonlinearity, and Complexity 14(3) (2025) 569--587 | DOI:10.5890/DNC.2025.09.010

Vasileios Drakopoulos$^1$, Song-Il Ri$^2$

$^1$ Department of Computer Science and Biomedical Informatics, University of Thessaly, Lamia, 35131, Greece

$^2$ Department of Mathematics, University of Science, Pyongyang, DPR Korea

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Abstract

Fractal interpolation offers a more flexible approach than traditional interpolation methods, thanks to techniques that use repeated transformations to build these unique functions. Fixed points -- the values that remain stable through these transformations -- are central to creating patterns that stay consistent, or `invariant', across different types of iterated function systems. Unlike classic methods like polynomial interpolation, fractal interpolation achieves its shape through a specialised operator, which ensures that the function holds its fractal form. A common foundation for this approach is the Banach fixed point theorem, which helps in reliably constructing these functions. This article reviews key methods for creating 2D fractal surfaces, focusing on techniques like Rakotch contractions and the Matkowski theorem, which expand the possibilities for fractal interpolation in real-world applications. It also includes some of the authors' recent findings. Notably, we build on past work with bivariable fractal interpolation functions to offer a more detailed perspective. The methods presented here diverge from existing techniques, offering straightforward ways to represent complicated patterns.

Acknowledgments

The authors greatly thank the anonymous referees for many valuable suggestions and comments which helped to improve the original paper.

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