Constrained univariate and bivariate rational fractal interpolation

Abstract

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 ${\mathcal{C}}^2.$ 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.

Keywords:

• Fractals
• iterated function systems
• splines
• rational fractal interpolation function
• convergence
• fractal interpolation surface
• constrained interpolation

Additional information

Funding

The second author is thankful for the project: MTR/2017/000574 - MATRICS from the Science and Engineering Research Board (SERB), Government of India. The authors are grateful to the anonymous reviewers for their valuable suggestions to improve the presentation of the paper.