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| By an appropriate selection of elements in the iterated function system, the corresponding fractal interpolation function can be tailor-made to represent differentiable or nondifferentiable functions. The notion of fractal interpolation can be effectively used to obtain an entire class of self-referential functions $f^\alpha$ parameterized by a certain vector quantity $\alpha$ with $f$ as its germ. For suitable values of the parameter, the fractal functions $f^\alpha$ simultaneously approximate and interpolate a prescribed continuous function $f$. The current article aims to identify appropriate values of the parameters so that the perturbed function $f^\alpha$ shares some properties such as regularity and positivity with the original function $f$. Consequently, elementary theorems on ``property preserving fractal perturbation process'' established in this article pave the way to shape preserving fractal approximation theory. Another problem exposed in this article is the approximation of a nonnegative function from below by nonnegative fractal splines. Further, some quantitative estimates in the copositive approximation of smooth functions by the fractal polynomials are broached. Fractal element introduced in the shape preserving approximant opens a wide range of choices for the selection of a uniform approximant corresponding to a given continuous function. Overall, the paper can be viewed as an earnest attempt to bridge the gap between the two fields that are developing in parallel - the theory of shape preserving approximation and fractal functions - for the one to benefit from the other. |
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