Journal of Inequalities and Applications 
Volume 2006 (2006), Article ID 78734, 20 pages
doi:10.1155/JIA/2006/78734

Smooth fractal interpolation

M. A. Navascués1 and M. V. Sebastián2

 1Departamento de Matemática Aplicada, Universidad de Zaragoza, C/María de Luna 3, Zaragoza 50018, Spain
2Departamento de Matemáticas, Universidad de Zaragoza, Campus Plaza de San Francisco s/n, Zaragoza 50009, Spain
Received 12 December 2005; Revised 5 May 2006; Accepted 14 June 2006

Abstract

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 ℋ2[a,b] is proven.