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| Fractal interpolation that possesses the ability to produce smooth and nonsmooth interpolants is a novice to the subject of interpolation. Apart from appropriate degree of smoothness, a good interpolant should reflect shape properties, for instance monotonicity, inherent in a prescribed data set. To address this requirement, several shape preserving interpolation schemes are developed in the literature. There are a few articles concerned with intersection of these two fields, i.e., shape preserving fractal interpolation initiated by our group. Despite the flexibility offered by these shape preserving fractal interpolants, they are well-suited only for the representation of self-referential functions. We introduce a new class of monotone $\mathcal{C}^1$-cubic interpolants by taking full advantage of flexibility offered by the hidden variable fractal interpolation functions (HVFIFs). We achieve this using a two-step procedure enunciated in the following. Firstly, we associate an entire family of $\mathbb{R}^2$-valued continuous functions $\mathbf{f}[\mathbf{A}]$ parameterized by a suitable block matrix $\mathbf{A}$ with a prescribed function $f \in \mathcal{C}(I, \mathbb{R}^2)$. Secondly, we impose appropriate constraints on $\mathbf{A}$ so that $\mathbf{f}[\mathbf{A}]$ preserves monotonicity inherent in $\textbf{f}$. This theory invoked to the $\mathcal{C}^1$-cubic spline HVFIF, which can be viewed as a fractal perturbation of the traditional
$\mathcal{C}^1$-cubic spline, culminate with the desired monotonicity preserving
$\mathcal{C}^1$-cubic HVFIF. The monotonicity preserving interpolation scheme developed herein generalizes and enriches its traditional nonrecursive counterpart and its fractal extension. |
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