A new class of rational cubic spline fractal interpolation function and its constrained aspects

摘要

This paper pertains to the area of shape preservation and sets a theoretical foundation for the applications of preserving constrained nature of a given constraining data in fractal interpolation functions (FIFs) techniques. We construct a new class of rational cubic spline FIFs (RCSFIFs) with a preassigned quadratic denominator with two shape parameters, which includes classical rational cubic interpolant [Appl. Math. Comp., 216 (2010), pp. 2036–2049] as special case and improves the sufficient conditions for positivity. Convergence analysis of RCSFIF to the original function in C1 is studied. In order to meet the needs of practical design or overcome the drawback of the tension effect in the proposed RCSFIFs, we improve our method by introducing a new tension parameter wi and construct a new class of rational cubic spline FIFs with three shape parameters. The scaling factors and shape parameters have a predictable adjusting role on the shape of curves. The elements of the rational iterated function system in each subinterval are identified befittingly so that the graph of the resulting C1-rational cubic spline FIF constrained (i) within a prescribed rectangle (ii) above a prescribed straight line (iii) between two piecewise straight lines. These parameters include, in particular, conditions on the positivity of the C1-rational cubic spline FIF. Several numerical examples are presented to ascertain the correctness and usability of developed scheme and to suggest how these schemes outperform their classical counterparts.