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 Wₙ(x, y) = (aₙ x+eₙ, αₙ(x) y +ψₙ(x)), n = 1, …, N. Then, we may define the generalized affine FIF f interpolating a given data set \ (xₙ, yₙ) ∈ I× R, n = 0, 1, …, N \, where I = [x₀, xN]. In this paper, we discuss the smoothness of the FIF f. We prove, in particular, that f is θ-hölder function whenever ψₙ are. Furthermore, we give the appropriate upper bound of the maximum range of FIF in this case.