Constrained fractal interpolation functions with variable scaling

Fractal interpolant function (FIF) constructed through iterated function systems is more general than classical spline interpolant. In this paper, we introduce a family of rational cubic splines with variable scaling, where the numerators and denominators of rational function are cubic and linear polynomial respectively. FIFs with variable scaling offer more flexibility in fitting and approximation of many complicated phenomena than that of in FIF with constant scaling. The convergence result of the proposed rational cubic interpolant to data generating function in $\mathcal{C}^1$ is proven. When interpolation data is constrained by piecewise curves, we derive sufficient condition on the parameter of rational FIF so that it lies between them.