Constrained univariate and bivariate rational fractal interpolation

We study constrained natures of a new class of univariate and bivariate rational cubic fractal interpolation functions (RCFIFs). We derive the convergence results of the RCFIF towards an original function in C-2. In particular, when data lies (i) between two piecewise defined lines (ii) within a rectangle, we derive sufficient condition based on the restrictions of IFS parameters at fewer discretized values so that the corresponding RCFIF preserves the inherent property associated with constrained data. Using transfinite interpolation via blending functions, we extend constrained aspects to rational bivariate RCFIFs that lie above a piecewise plane and within a cuboid.