On the construction of recurrent fractal interpolation functions using Geraghty contractions

Najmeddine Attia; Hajer Jebali; Najmeddine Attia; Hajer Jebali

doi:10.3934/era.2023347

Electronic Research Archive

2023, Volume 31, Issue 11: 6866-6880. doi: 10.3934/era.2023347

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Research article

On the construction of recurrent fractal interpolation functions using Geraghty contractions

Najmeddine Attia ^{1,2
,
,},
Hajer Jebali ²

1.
Department of Mathematics and Statistics, College of Science, King Faisal University, PO. Box: 400 Al-Ahsa 31982, Saudi Arabia
2.
Analysis, Probability and Fractals Laboratory LR18ES17, Department of Mathematics, Faculty of Sciences of Monastir, University of Monastir, 5000-Monastir, Tunisia

Received: 21 July 2023 Revised: 22 September 2023 Accepted: 07 October 2023 Published: 26 October 2023

The recurrent iterated function systems (RIFS) were first introduced by Barnsley and Demko and generalized the usual iterated function systems (IFS). This new method allowed the construction of more general sets, which do not have to exhibit the strict self similarity of the IFS case and, in particular, the construction of recurrent fractal interpolation functions (RFIF). Given a data set $ \{ (x_n, y_n) \in I\times \mathbb R, n = 0, 1, \ldots, N \} $ where $ I = [x_0, x_N] $, we ensured that attractors of RIFS constructed using Geraghty contractions were graphs of some continuous functions which interpolated the given data. Our approach goes beyond the classical framework and provided a wide variety of systems for different approximations problems and, thus, gives more flexibility and applicability of the fractal interpolation method. As an application, we studied the error rates of time series related to the vaccination of COVID-19 using RFIF, and we compared them with the obtained results on the FIF.
- recurrent iterated function system (RIFS),
- Geraghty contraction,
- recurrent fractal interpolation function
Citation: Najmeddine Attia, Hajer Jebali. On the construction of recurrent fractal interpolation functions using Geraghty contractions[J]. Electronic Research Archive, 2023, 31(11): 6866-6880. doi: 10.3934/era.2023347

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Abstract

The recurrent iterated function systems (RIFS) were first introduced by Barnsley and Demko and generalized the usual iterated function systems (IFS). This new method allowed the construction of more general sets, which do not have to exhibit the strict self similarity of the IFS case and, in particular, the construction of recurrent fractal interpolation functions (RFIF). Given a data set $ \{ (x_n, y_n) \in I\times \mathbb R, n = 0, 1, \ldots, N \} $ where $ I = [x_0, x_N] $, we ensured that attractors of RIFS constructed using Geraghty contractions were graphs of some continuous functions which interpolated the given data. Our approach goes beyond the classical framework and provided a wide variety of systems for different approximations problems and, thus, gives more flexibility and applicability of the fractal interpolation method. As an application, we studied the error rates of time series related to the vaccination of COVID-19 using RFIF, and we compared them with the obtained results on the FIF.

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