Hidden-variable alpha-fractal functions and their monotonicity aspects.

Fractal interpolation that possesses the ability to produce smooth and nonsmooth inter- polants is a novice to the subject of interpolation. Apart from appropriate degree of smooth- ness, a good interpolant should reflect shape properties, for instance monotonicity, inherent in a prescribed data set. Despite the flexibility offered by these shape preserving fractal interpolants developed recently in the literature are well-suited only for the representation of self-referential functions. In this article we present hidden variable A-fractal interpo- lation function as a tool to associate an entire family of R2-valued continuous functions f[A] parameterized by a suitable block matrix A with a prescribed function f ¿ C(I,R2). Depending on the choice of parameters, the members of the family may be self-referential, or non-self-referential, and preserve some properties of original function f, thus yielding more diversity and flexibility in the process of approximation. As an application of the developed theory, we introduce a new class of monotone C1-cubic interpolants by taking full advantage of flexibility offered by the hidden variable A-fractal interpolation functions (HFIFs). This theory invoked to the C1-cubic spline HFIF, which can be viewed as a fractal perturbation of the traditional C1-cubic spline, culminates with the desired monotonicity preserving C1-cubic HFIF. The monotonicity preserving interpolation scheme developed herein generalizes and enriches its traditional nonrecursive counterpart and its fractal ex- tension