Fractal Approximation

Abstract

In the present article every complex square integrable function defined in a real bounded interval is approached by means of a complex fractal function. The approximation depends on a partition of the interval and a vectorial parameter of the iterated function system providing the fractal attractor. The original may be discontinuous or undefined in a set of zero measure. The fractal elements can modify the features of the originals, for instance their character of smooth or non-smooth. The properties of the operator mapping every function into its fractal analogue are studied in the context of the uniform and least square norms. In particular, the transformation provides a decomposition of the set of square integrable maps. An orthogonal system of fractal functions is constructed explicitly for this space. Sufficient conditions for the uniform convergence of the fractal series expansion corresponding to this basis are also deduced. The fractal approximation of real functions is obtained as a particular case.

Fractal Bases for Banach Spaces of Smooth functions

“…although, as proved in [8], all the results obtained can be generalised to the complex field. For any r ∈ N, let N r = {1, 2, .…”

Toward a Unified Methodology for Fractal Extension of Various Shape Preserving Spline Interpolants

“…Step 4: Input the scaling parameters as prescribed by Step 3 in the functional equation represented by (12) whereupon the points of the graph of f α are computed.…”

“…Further, differentiable FIFs [2] constitute an alternative to the traditional nonrecursive interpolation and approximation methods (see, for instance, [3,5,12]). In this way, the fractal methodology provides more flexibility and versatility on the choice of an interpolant.…”

Fractal Functions in Interpolation and Approximation: A Bird's-Eye View

“…Following the publication of Fractals Everywhere [2], a beautiful exposition of IFS theory, fractal functions and their applications, various related issues such as calculus, Holder continuity, convergence, stability, smoothness, determination of scaling parameters, and perturbation error have been investigated in the literature [3][4][5][6][7][8][9][10][11][12][13]. The concept of smooth FIFs has been used to generalize the traditional splines [14][15][16][17][18] and to demonstrate that the interaction of classical numerical methods with fractal theory provides new interpolation schemes that supplement the existing ones. Various other extensions of FIFs include multivariable FIFs [7,9,16,[19][20][21][22][23][24][25] generated by using higher-dimensional or recurrent IFSs, the hidden variable FIFs produced by projecting the attractors of vector valued IFSs to a lower-dimensional space.…”

“…These maps tend to bridge the gap between smoothness of classical mathematical objets and pseudo-randomness of experimental variables, breaking in this way their apparent diversity. Navascues and coworkers [17,18,22] contributed to the theory by defining "rough" approximants as perturbation of the functions generally used in classical approximation (polynomial, trigonometric, rational, etc.) via this operator.…”