On the Fractal interpolation functions associated with Matkowski contractions

Attia, Najmeddine; Balegh, Mohamed; Amami, Rim

doi:10.3934/era.2023238

era

2023

DOI: 10.3934/era.2023238

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On the Fractal interpolation functions associated with Matkowski contractions

Najmeddine Attia

Mohamed Balegh

Rim Amami

Abstract: <abstract>In this paper we investigate an iterated function system that defines a fractal interpolation function, where ordinate scaling, that is Lipschitz constant in Banach contraction principle is substituted by real-valued control function. In such a manner, fractal interpolation functions associated with Matkowski contractions are obtained and provide a new framework of approximating experimental data. Furthermore, given a data generating function $ f $, we study a new class of fractal inte… Show more

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Cited by 4 publications

References 32 publications

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On the construction of recurrent fractal interpolation functions using Geraghty contractions

Attia,

Jebali

2023

era

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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.</abstract>

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On the construction of recurrent fractal interpolation functions using Geraghty contractions

Attia,

Jebali

2023

era

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Note on fractal interpolation function with variable parameters

Attia,

Moulahi,

Amami

et al. 2023

MATH

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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.</abstract>

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On the stability of Fractal interpolation functions with variable parameters

Attia,

Saidi,

Amami

et al. 2024

MATH

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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.</abstract>

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On the Fractal interpolation functions associated with Matkowski contractions

Cited by 4 publications

References 32 publications

On the construction of recurrent fractal interpolation functions using Geraghty contractions

On the construction of recurrent fractal interpolation functions using Geraghty contractions

Note on fractal interpolation function with variable parameters

On the stability of Fractal interpolation functions with variable parameters

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