Note on fractal interpolation function with variable parameters - Attia, Najmeddine; Moulahi, Taoufik; Amami, Rim; Saidi, Neji (AIMS mathematics, 2024)
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, α_n(x) y +ψ_n(x)\big) $, $ n = 1, …, N $. Then, we may define the generalized affine FIF $ f $ interpolating a given data set $ \big\{ (x_n, y_n) \in I× \mathbb R, n = 0, 1, …, 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 $ θ $-hölder function whenever $ ψ_n $ are. Furthermore, we give the appropriate upper bound of the maximum range of FIF in this case.