 Applied Mathematics, 2010, 1, 200-210 doi:10.4236/am.2010.13024 Published Online September 2010 (http://www.SciRP.org/journal/am) Copyright © 2010 SciRes. AM On Complete Bicubic Fractal Splines Arya Kumar Bedabrata Chand1, María Antonia Navascués2 1Department of Mathematics, Indian Institute of Technology Madras, Chennai, India 2Departmento de Matemática Aplicada, Centro Politécnico Superior de Ingenieros, Universidad de Zaragoza, Zaragoza, Spain E-mail: chand@iitm.ac.in, manavas@unizar.es Received June 6, 2010; revised July 20, 2010; accepted July 23, 2010 Abstract Fractal geometry provides a new insight to the approximation and modelling of experimental data. We give the construction of complete cubic fractal splines from a suitable basis and their error bounds with the origi-nal function. These univariate properties are then used to investigate complete bicubic fractal splines over a rectangle . Bicubic fractal splines are invariant in all scales and they generalize classical bicubic splines. Finally, for an original function 4[]fC, upper bounds of the error for the complete bicubic fractal splines and derivatives are deduced. The effect of equal and non-equal scaling vectors on complete bicubic fractal splines were illustrated with suitably chosen examples. Keywords: Fractals, Iterated Function Systems, Fractal Interpolation Functions, Fractal Splines, Surface Approximation. 1. Introduction Schoenberg  introduced “spline functions” to the mathematical literature. In the last 60 years, splines have proved to be enormously important in different branches of mathematics such as numerical analysis, numerical treatment of differential, integral and partial differential equations, approximation theory and statistics. Also, splines play major roles in field of applications, such as CAGD, tomography, surgery, animation and manufac-turing. In this paper, we discuss on complete fractal splines that generalize the classical complete splines. Fractal interpolation functions (FIFs) were introduced by Barnsley [2,3] based on the theory of iterated function system (IFS). The attractor of the IFS is the graph of FIF that interpolates a given set of data points. Fractal inter-polation constitutes an advance in the sense that the functions used are not necessarily differentiable and show the rough aspect of real-world signals [3,4]. A spe-cific feature is the fact that the graph of these interpo-lants possesses a fractal dimension and this parameter provides a geometric characterization of the measured variable which may be used as an index of the complex-ity of a phenomenon. Barnsley and Harrington  first constructed a differentiable FIF or rC-FIF f that interpolates the prescribed data if values of ()kf, =1,2,, ,kr at the initial end point of the interval are given. In this construction, specifying boundary condi-tions similar to those of classical splines was found to be quite difficult to handle. The fractal splines with general boundary conditions are studied recently [6,7]. The power of fractal methodology allows us to generalize almost any other interpolation techniques, see for in-stance [8-10]. Fractal surfaces have proved to be useful functions in scientific applications such as metallurgy, physics, geol-ogy, image processing and computer graphics. Mas-sopust  was first to put forward the construction of fractal interpolation surfaces (FISs) on triangular do-mains, where the interpolation points on the boundary of the domain are coplanar. Geronimo and Hardin , and Zhao  generalized this construction in different ways. The general bivariate FIS on rectangular grids are treated for instance in references [14,15]. Recently, Bouboulis and Dalla constructed fractal interpolation surfaces from FIFs through recurrent iterated function systems . In this paper we approach the problem of complete cubic spline surface from a fractal perspective. In Section 2, we construct cardinal cubic fractal splines through moments and estimate the error bound of the complete cubic spline with the original function. The construction of bicubic fractal splines is carried out in Section 3 through tensor products. Finally, for an original function 4[]fC, upper bounds of the error for the complete A. K. B. CHAND ET AL. Copyright © 2010 SciRes. AM 201bicubic fractal splines and derivatives are deduced. The effect of scaling factors on bicubic fractal splines are demonstrated in the last section through various exam-ples. 2. Complete Cubic Fractal Splines We discuss on fractal interpolation based on IFS theory in Subsection 2.1 and construct cardinal cubic fractal spline through moments in Subsection 2.2. Upper bounds of L-norm of the error of a complete cubic spline FIF with respect to the original function are deduced in Sub-section 2.3. 2.1. Fractal Interpolation Functions Let 01:<< ...