On linear transformation of generalized affine fractal interpolation function

Najmeddine Attia; Rim Amami; Najmeddine Attia; Rim Amami

doi:10.3934/math.2024817

AIMS Mathematics

2024, Volume 9, Issue 7: 16848-16862. doi: 10.3934/math.2024817

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Research article

On linear transformation of generalized affine fractal interpolation function

Najmeddine Attia ^{1,2
,
,},
Rim Amami ³

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

Received: 19 March 2024 Revised: 24 April 2024 Accepted: 06 May 2024 Published: 14 May 2024
MSC : 28A80, 47H10, 65D05

In this work, we investigate a class of generalized affine fractal interpolation functions (FIF) with variable parameters, where ordinate scaling is substituted by a real-valued control function. Let $ {\mathcal S} $ be an iterated function system (IFS) with the attractor $ G_\Delta $, where $ \Delta $ is a given data set. We consider an affine transformation $ \omega(\Delta) $ of $ \Delta $, and we define the IFS $ \hat {\mathcal S} $ with the attractor $ G_{\omega(\Delta)} $. We give a sufficient condition so that $ G_{\omega(\Delta)} = \omega(G_\Delta) $. In addition, we compare the definite integrals of the corresponding FIF and study the additivity property. Some examples will be given, highlighting the effectiveness of our results.
- iterated function system,
- generalized affine fractal interpolation function,
- linear transformation
Citation: Najmeddine Attia, Rim Amami. On linear transformation of generalized affine fractal interpolation function[J]. AIMS Mathematics, 2024, 9(7): 16848-16862. doi: 10.3934/math.2024817

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Abstract

In this work, we investigate a class of generalized affine fractal interpolation functions (FIF) with variable parameters, where ordinate scaling is substituted by a real-valued control function. Let $ {\mathcal S} $ be an iterated function system (IFS) with the attractor $ G_\Delta $, where $ \Delta $ is a given data set. We consider an affine transformation $ \omega(\Delta) $ of $ \Delta $, and we define the IFS $ \hat {\mathcal S} $ with the attractor $ G_{\omega(\Delta)} $. We give a sufficient condition so that $ G_{\omega(\Delta)} = \omega(G_\Delta) $. In addition, we compare the definite integrals of the corresponding FIF and study the additivity property. Some examples will be given, highlighting the effectiveness of our results.

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