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The fractal interpolation functions with appropriate iterated function systems provide a method to perturb and approximate a continuous function on a compact interval I. This method produces a class of functions f ∈ C(I) , where is a vector with functional components. The presence of scaling function in these fractal functions helps to get a wide variety of mappings for approximation problems. The current article explores the shape-preserving properties of the -fractal functions with variable scalings, where the optimal ranges of the scaling functions are derived for fundamental shapes of the germ f. We provide several examples to illustrate the shape preserving results and apply our fractal methodologies in approximation problems. Also, it is shown that the order of convergence of the -fractal polynomial to the original shaped function matches with that of polynomial approximation. Further, based on the shape preserving properties of the -fractal functions, we provide the fractal analogue of the Chebyshev alternation theorem. To the end, we deduce the fractal version of the classical full Müntz theorem in C[0, 1].

This article proposes a generalization of the Fourier interpolation formula, where a wider range of the basic trigonometric functions is considered. The extension of the procedure is done in two ways: adding an exponent to the maps involved, and considering a family of fractal functions that contains the standard case. The studied interpolation converges for every continuous function, for a large range of the nodal mappings chosen. The error of interpolation is bounded in two ways: one theorem studies the convergence for Hölder continuous functions and other develops the case of merely continuous maps. The stability of the approximation procedure is proved as well.

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