A revisit to smoothness preserving fractal perturbation of a bivariate function: Self-Referential counterpart to bicubic splines

Abstract

Construction of fractal interpolation surfaces has recently been considered in the standpoint of a parameterized class of fractal (self-referential) functions corresponding to a given bivariate continuous function. In this paper, we describe a procedure so that the elements in this parameterized class preserve smoothness (C(2,2)-regularity) of the original bivariate function defined on a rectangle. As a consequence, we generalize the bicubic spline by means of a two-parameter family of fractal functions, which we call bicubic fractal splines. Under certain hypotheses, upper bounds for the interpolation error for the bicubic fractal spline and its derivatives are obtained. A detailed exposition of C(2,2)-regular self-referential functions is provided not only as a prelude to the bicubic fractal splines, but also to elucidate the study of smoothness preserving bivariate self-referential functions appeared recently in the fractal literature.