Multivariate fractal interpolation functions: some approximation aspects and an associated fractal interpolation operator

Abstract

In the classical (non-fractal) setting, the natural kinship between theories of interpolation and approximation is well explored. In contrast to this, in the context of fractal interpolation, the interrelation between interpolation and approximation is subtle, and this duality is relatively obscure. The notion of α-fractal functions provides a proper foundation for the approximation-theoretic facet of univariate fractal interpolation functions (FIFs). However, no comparable approximation-theoretic aspects of FIFs have been developed for functions of several variables. The current article intends to open the door for intriguing interactions between approximation theory and multivariate FIFs. To this end, in the first part of this article, we develop a general framework for constructing multivariate FIFs, which is amenable to provide a multivariate analogue of an α-fractal function. Multivariate α-fractal functions provide a parameterized family of fractal approximants associated with a given multivariate continuous function. Some elementary aspects of the multivariate fractal (not necessarily linear) interpolation operator that sends a continuous function defined on a hyperrectangle to its fractal analogue are studied. As in the univariate setting, the notion of α-fractal functions serves as a basis for fractalizing various results in multivariate approximation theory, including that of multivariate splines. For our part, we provide some approximation classes of multivariate fractal functions and prove a few results on the constrained fractal approximation of real-valued continuous functions of several variables.

Multivariate Zipper Fractal Functions

“…Based on the Hölder exponents of germ function and base function, and scaling factors, we derive the bounds for box-dimension of multivariate zipper α-fractal function. Our results are more general than several existing results in univariate and multivariate cases [1,32,39].…”

“…, m. Then, we can obtain 2 m -multivariate FIFs by zipper methodology for the same set of scaling. When all = 0, then our multivariate zipper fractal function reduces to the existing simple multivariate fractal function [32].…”

Contractive Multivariate Zipper Fractal Interpolation Functions

“…In this section we work under the following supplementary conditions which are natural in view of [16,20,23]: (α)…”