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 $W_n(x,y)=(a_{n}x+e_n,{\alpha }_n(x)y+{\psi }_n(x))$, $n=1,\dots ,N$. Then, we may define the generalized affine FIF $f$ interpolating a given data set $\{(x_n,y_n)\in I\times \mathbb{R},n=0,1,\dots ,N\}$, 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.