Towards a More General Type of Univariate Constrained Interpolation With Fractal Splines
For researchers in fractal interpolation, this provides a more flexible constrained interpolation method, but the contribution is incremental as it generalizes existing techniques without introducing a new paradigm.
This paper develops a more general method for univariate constrained interpolation using rational cubic fractal splines, ensuring interpolating curves lie strictly above or below prescribed linear or quadratic spline functions. The approach extends prior work by enabling constraints with arbitrary linear or quadratic splines.
Recently, in [Electronic Transaction on Numerical Analysis, 41 (2014), pp. 420-442] authors introduced a new class of rational cubic fractal interpolation functions with linear denominators via fractal perturbation of traditional nonrecursive rational cubic splines and investigated their basic shape preserving properties. The main goal of the current article is to embark on univariate constrained fractal interpolation that is more general than what was considered so far. To this end, we propose some strategies for selecting the parameters of the rational fractal spline so that the interpolating curves lie strictly above or below a prescribed linear or a quadratic spline function. Approximation property of the proposed rational cubic fractal spine is broached by using the Peano kernel theorem as an interlude. The paper also provides an illustration of background theory, veined by examples.