Note on fractal interpolation function with variable parameters

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 $. The...

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Published inAIMS mathematics Vol. 9; no. 2; pp. 2584 - 2601
Main Authors Attia, Najmeddine, Moulahi, Taoufik, Amami, Rim, Saidi, Neji
Format Journal Article
LanguageEnglish
Published AIMS Press 01.01.2024
Subjects
generalized affine fractal interpolation function
hölder and lipschitz functions
iterated function system
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ISSN2473-6988
2473-6988
DOI10.3934/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.
AbstractList 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.
Author Attia, Najmeddine
Moulahi, Taoufik
Amami, Rim
Saidi, Neji
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Snippet 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...
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SubjectTerms generalized affine fractal interpolation function
hölder and lipschitz functions
iterated function system
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Title Note on fractal interpolation function with variable parameters
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