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Bernstein super fractal interpolation function for countable data systems

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Abstract

We introduce a fractal operator on \(\mathcal {C}[0,1]\) which sends a function \(f \in \mathcal {C}(I)\) to fractal version of f where fractal version of f is a super fractal interpolation function corresponding to a countable data system. Furthermore, we study the continuous dependence of super fractal interpolation functions on the parameters used in the construction. We know that the invariant subspace problem and the existence of a Schauder basis gained lots of attention in the literature. Here, we also show the existence of non-trivial closed invariant subspace of the super fractal operator and the existence of fractal Schauder basis for \(\mathcal {C}(I)\). Moreover, we can see the effect of the composition of Riemann-Liouville integral operator and super fractal operator on the fractal dimension of continuous functions. We also mention some new problems for further investigation.

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Acknowledgements

We are thankful to the anonymous reviewers and editor for their constructive comments and suggestions, which helped us to improve the manuscript considerably.

Funding

The work of the first author (SC) is financially supported by the CSIR, India, with fellowship number 09/1058(0012)/2018-EMR-I. The third author (SV) thanks Prof. Syed Abbas for providing financial support to carry out this work.

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Authors and Affiliations

  1. School of Basic Sciences, Indian Institute of Technology Mandi, Kamand (H.P.), 175005, India

    Subhash Chandra & Syed Abbas

  2. Department of Applied Sciences, IIIT Allahabad, Prayagraj, 211015, India

    Saurabh Verma

Authors
  1. Subhash Chandra
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  2. Syed Abbas
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  3. Saurabh Verma
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Correspondence to Syed Abbas.

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Chandra, S., Abbas, S. & Verma, S. Bernstein super fractal interpolation function for countable data systems. Numer Algor 92, 2457–2481 (2023). https://doi.org/10.1007/s11075-022-01398-5

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