Smooth fractal interpolation
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Fractal methodology provides a general frame for the understanding of real-world phenomena. In particular, the classical methods of real-data interpolation can be generalized by means of fractal techniques. In this paper, we describe a procedure for the construction of smooth fractal functions, with the help of Hermite osculatory polynomials. As a consequence of the process, we generalize any smooth interpolant by means of a family of fractal functions. In particular, the elements of the class can be defined so that the smoothness of the original is preserved. Under some hypotheses, bounds of the interpolation error for function and derivatives are obtained. A set of interpolating mappings associated to a cubic spline is defined and the density of fractal cubic splines in Ᏼ2 [a,b] is proven. Copyright © 2006 M. A. Navascu´es and M. V. Sebasti´an. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 1. Introduction Fractal interpolation techniques provide good deterministic representations of complex phenomena. Barnsley [2, 3] and Hutchinson [8] were pioneers in the use of fractal functions to interpolate sets of data. Fractal interpolants can be defined for any continuous function defined on a real compact interval. This method constitutes an advance in the techniques of approximation, since all the classical methods of real-data interpolation can be generalized by means of fractal techniques (see, e.g., [5, 10, 12]). Fractal interpolation functions are defined as fixed points of maps between spaces of functions using iterated function systems. The theorem of Barnsley and Harrington (see [4]) proves the existence of differentiable fractal interpolation functions. However, in some cases, it is difficult to find an iterated funcion system satisfying the hypotheses of the theorem, mainly whenever some specific boundary conditions are required (see [4]). In this paper, we describe a very general way of constructing smooth fractal Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2006, Article ID 78734, Pages 1–20 DOI 10.1155/JIA/2006/78734
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Smooth fractal interpolation
functions with the help of Hermite osculatory polynomials. The proposed method solves the problem with the help of a classical interpolant. The fractal solution is unique and the constructed interpolant preserves the prefixed boundary conditions. The procedure has a computational cost similar to that of the classical method. As a consequence of the process, we generalize any smooth interpolant by means of a family of fractal functions. Each element of the class preserves the smoothness and the boundary conditions of the original. Under some hypotheses, bounds of the interpolation error for function and derivatives are obtained. Assuming some additional conditions on the scaling factors, the convergence is also preserved. In the last section, a set of interpolating ma
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