A revisit to smoothness preserving fractal perturbation of a bivariate function: Self-Referential counterpart to bicubic splines. (April 2022) - Viswanathan, P.
Highlights: Revisited smoothness preserving bivariate α -fractal functions to clarify and correct construction scattered in a couple of recent literature. This improved construction is applied to introduce and investigate bicubic fractal splines. Space of bicubic fractal splines is introduced using fractal operator. An upper bound for approximation error is established. 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.