Note on fractal interpolation function with variable parameters

• Published in: AIMS mathematics Vol. 9; no. 2; pp. 2584 - 2601
• Main Authors: Attia, Najmeddine, Moulahi, Taoufik, Amami, Rim, Saidi, Neji
• Format: Journal Article
• Language: English
• Published: AIMS Press 01.01.2024
• Subjects: generalized affine fractal interpolation function; hölder and lipschitz functions; iterated function system
• ISSN: 2473-6988; 2473-6988
• DOI: 10.3934/math.2024127

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.