Article ID: MTJPAM-D-26-00074

Title: Complex fractal interpolation on 1-chains


Montes Taurus J. Pure Appl. Math. / ISSN: 2687-4814

Article ID: MTJPAM-D-26-00074; Volume 8 / Issue 2 / Year 2026, Pages 71-83

Document Type: Research Paper

Author(s): Peter Massopust a

aDepartment of Mathematics, Technical University of Munich, Boltzmannstrasse 3, 85748 Garching by Munich, Germany

Received: 21 April 2026, Accepted: 20 June 2026, Published: 25 October 2026

Corresponding Author: Peter Massopust (Email address: massopust@ma.tum.de)

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Abstract

In this paper, we introduce the novel concept of fractal interpolation on 1-chains in the complex plane. The approach presented here extends to a certain degree the results obtained by the author for curves in Banach spaces (cf. [18]). We work within the framework of iterated function systems and fractal functions and employ their fundamental properties which are mainly based on the Banach fixed point theorem. In particular, we establish the existence of continuous and Lp fractal interpolation functions defined on 1-chains and present some examples. In addition, we derive an associated fractal operator for a space of continuous functions on 1-chains and discuss several of its properties.

Keywords: 1-chain, iterated function system, fractal interpolation, Read-Bajraktarević operator

References:
  1. M. F. Barnsley, Fractal functions and interpolation, Constr. Approx. 2, 303–329, 1986.
  2. M. F. Barnsley, Fractals everywhere, Academic Press, Orlando, 1988.
  3. M. F. Barnsley, M. Hegland and P. R. Massopust, Self-referential functions 2016; ArXiv: 1610.01369.
  4. M. F. Barnsley, J. Elton, D. Hardin and P. R. Massopust, Hidden variable fractal interpolation functions, SIAM J. Math. Anal. 20 (5), 1218–1242, 1989.
  5. M. F. Barnsley, B. Harding, C. Vince and P. Viswanathan, Approximation of rough functions, J. Approx. Theory 209, 23–43, 2016.
  6. T. Bedford, M. Dekking and M. Keane, Fractal image coding techniques and contraction operators, Tech. Report 92-93, Delft University of Technology, Delft, Netherlands, 1992.
  7. E. Čech, Point sets, Academia, Publishing House of the Czechoslovak Academy of Sciences, Prague, 1969.
  8. S. Dubuc, Interpolation through an iterative scheme, J. Math. Anal. Appl. 114 (1), 185–204, 1986.
  9. A. Hatcher, Algebraic topology, Cambridge University Press, London, 2002.
  10. F. Hausdorff, Grundzüge der Mengenlehre, Viet, Leipzig, 1914.
  11. J. E. Hutchinson, Fractals and self-similarity, Indiana Univ. Math. J. 30 (5), 713–747, 1981.
  12. C. Loeh, Isomorphisms in ℓ1-homology, Münster J. Math. 1, 237–266, 2008.
  13. P. R. Massopust, Interpolation and approximation with splines and fractals, Oxford University Press, Oxford, 2010.
  14. P. R. Massopust, On local fractal functions in Besov and Triebel–Lizorkin spaces, J. Math. Anal. Appl. 436, 393–407, 2016.
  15. P. R. Massopust, Fractal functions, fractal surfaces, and wavelets (Second Edition), Academic Press, San Diego, 2016.
  16. P. R. Massopust, Fractal interpolation over nonlinear partitions, Chaos, Solitons & Fractals 162, 1–8, 2022; Article ID: 112503, https://doi.org/10.1016/j.chaos.2022.112503.
  17. P. R. Massopust, Fractal hypersurfaces, affine Weyl groups, and wavelet sets, J. Anal. 32, 399–431, 2024; https://doi.org/10.1007/s41478-023-00653-9.
  18. P. R. Massopust, Fractal interpolation over curves, Contemporary Math. 797, 61–73, 2024.
  19. M. A. Navascués, Fractal polynomial interpolation, Z. Anal. Anwend. 24 (2), 401–418, 2005.
  20. M. A. Navascués and A. K. B. Chand, Fundamental sets of fractal functions, Acta Appl. Math. 100, 247–261, 2008.
  21. R. Pasupathi and R. Miculescu, A very general framework for fractal interpolation functions, J. Math. Anal. Appl. 534 (2), 1–17, 2024; Article ID: 128093.
  22. E. Peschl, Funktionentheorie – Band 1, BI Hochschultaschenbücher, Band 131, Mannheim, 1967.
  23. P. Renteln, Manifolds, tensors, and forms, Cambridge University Press, Cambridge, 2014.
  24. S. Rolewicz, Metric linear spaces, D. Reidel Publishing Company, Dordrecht, 1985.
  25. C. Serpa and J. Buesca, Constructive solutions for systems of iterative functional equations, Constr. Approx. 45 (2), 273–299, 2017.

Cite this article

How to cite this article: P. Massopust, Complex fractal interpolation on 1-chains, Montes Taurus J. Pure Appl. Math. 8 (2), 71-83, 2026; Article ID: MTJPAM-D-26-00074.