On the stability of Fractal interpolation functions with variable parameters

Najmeddine Attia; Neji Saidi; Rim Amami; Rimah Amami; Najmeddine Attia; Neji Saidi; Rim Amami; Rimah Amami

doi:10.3934/math.2024143

AIMS Mathematics

2024, Volume 9, Issue 2: 2908-2924. doi: 10.3934/math.2024143

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Research article

On the stability of Fractal interpolation functions with variable parameters

1.
Department of Mathematics and Statistics, College of Science, King Faisal University, P.O. Box 400, Al-Ahsa 31982, Saudi Arabia
2.
Department of Basic Sciences, Deanship of Preparatory Year and Supporting Studies, Imam Abdulrahman Bin Faisal University, P.O. Box 1982, Dammam 34212, Saudi Arabia
3.
Computer Department, Deanship of Preparatory Year and Supporting Studies, Imam Abdulrahman Bin Faisal University, P.O. Box 1982, Dammam 34212, Saudi Arabia

Received: 05 November 2023 Revised: 22 December 2023 Accepted: 27 December 2023 Published: 02 January 2024
MSC : 28A80, 47H10, 54H25, 37C70

Fractal interpolation function (FIF) is a fixed point of the Read–Bajraktarević operator defined on a suitable function space and is constructed via an iterated function system (IFS). In this paper, we considered the generalized affine FIF generated through the IFS defined by the functions $ W_n(x, y) = \big(a_n(x)+e_n, \alpha_n(x) y +\psi_n(x)\big) $, $ n = 1, \ldots, N $. We studied the shift of the fractal interpolation curve, by computing the error estimate in response to a small perturbation on $ \alpha_n(x) $. In addition, we gave a sufficient condition on the perturbed IFS so that it satisfies the continuity condition. As an application, we computed an upper bound of the maximum range of the perturbed FIF.
- iterated function system (IFS),
- generalized affine fractal interpolation function,
- hölder and Lipschitz functions
Citation: Najmeddine Attia, Neji Saidi, Rim Amami, Rimah Amami. On the stability of Fractal interpolation functions with variable parameters[J]. AIMS Mathematics, 2024, 9(2): 2908-2924. doi: 10.3934/math.2024143

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Abstract

Fractal interpolation function (FIF) is a fixed point of the Read–Bajraktarević operator defined on a suitable function space and is constructed via an iterated function system (IFS). In this paper, we considered the generalized affine FIF generated through the IFS defined by the functions $ W_n(x, y) = \big(a_n(x)+e_n, \alpha_n(x) y +\psi_n(x)\big) $, $ n = 1, \ldots, N $. We studied the shift of the fractal interpolation curve, by computing the error estimate in response to a small perturbation on $ \alpha_n(x) $. In addition, we gave a sufficient condition on the perturbed IFS so that it satisfies the continuity condition. As an application, we computed an upper bound of the maximum range of the perturbed FIF.

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