Note on fractal interpolation function with variable parameters

Najmeddine Attia; Taoufik Moulahi; Rim Amami; Neji Saidi; Najmeddine Attia; Taoufik Moulahi; Rim Amami; Neji Saidi

doi:10.3934/math.2024127

AIMS Mathematics

2024, Volume 9, Issue 2: 2584-2601. doi: 10.3934/math.2024127

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Research article

Note on fractal interpolation function 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 Mathematics, College of Science and Humanities in Al-Kharj, Prince Sattam Bin Abdulaziz University, Box 173 Al-Kharj 11942, Saudi Arabia
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: 31 October 2023 Revised: 04 December 2023 Accepted: 20 December 2023 Published: 26 December 2023
MSC : 28A80, 47H10, 65D05

Fractal interpolation function (FIF) is a new method of constructing new data points within the range of a discrete set of known data points. Consider the iterated functional system defined through the functions $ W_n(x, y) = \big(a_n x+e_n, \alpha_n(x) y +\psi_n(x)\big) $, $ n = 1, \ldots, N $. Then, we may define the generalized affine FIF $ f $ interpolating a given data set $ \big\{ (x_n, y_n) \in I\times \mathbb R, n = 0, 1, \ldots, N \big\} $, where $ I = [x_0, x_N] $. In this paper, we discuss the smoothness of the FIF $ f $. We prove, in particular, that $ f $ is $ \theta $-hölder function whenever $ \psi_n $ are. Furthermore, we give the appropriate upper bound of the maximum range of FIF in this case.
- iterated function system,
- generalized affine fractal interpolation function,
- hölder and Lipschitz functions
Citation: Najmeddine Attia, Taoufik Moulahi, Rim Amami, Neji Saidi. Note on fractal interpolation function with variable parameters[J]. AIMS Mathematics, 2024, 9(2): 2584-2601. doi: 10.3934/math.2024127

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Abstract

Fractal interpolation function (FIF) is a new method of constructing new data points within the range of a discrete set of known data points. Consider the iterated functional system defined through the functions $ W_n(x, y) = \big(a_n x+e_n, \alpha_n(x) y +\psi_n(x)\big) $, $ n = 1, \ldots, N $. Then, we may define the generalized affine FIF $ f $ interpolating a given data set $ \big\{ (x_n, y_n) \in I\times \mathbb R, n = 0, 1, \ldots, N \big\} $, where $ I = [x_0, x_N] $. In this paper, we discuss the smoothness of the FIF $ f $. We prove, in particular, that $ f $ is $ \theta $-hölder function whenever $ \psi_n $ are. Furthermore, we give the appropriate upper bound of the maximum range of FIF in this case.

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