A new class of rational cubic spline fractal interpolation function and its constrained aspects. (English) Zbl 1428.28016
Summary: 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 [M. Sarfraz et al. [ibid. 216, No. 7, 2036–2049 (2010; Zbl 1192.65028)] as special case and improves the sufficient conditions for positivity. Convergence analysis of RCSFIF to the original function in \(\mathcal{C}^1\) 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 \(w_i\) 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 \(\mathcal{C}^1\)-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 \(\mathcal{C}^1\)-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.
MSC:
| 28A80 | Fractals |
| 26A48 | Monotonic functions, generalizations |
| 41A05 | Interpolation in approximation theory |
| 41A20 | Approximation by rational functions |
| 65D05 | Numerical interpolation |
| 65D07 | Numerical computation using splines |
Keywords:
fractals; iterated function system; fractal interpolation functions; rational cubic fractal functions; rational cubic interpolation; positivityCitations:
Zbl 1192.65028Software:
pchipReferences:
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