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

TL;DR

This paper develops a framework for multivariate fractal interpolation functions, introducing multivariate $oldsymbol{ extalpha}$-fractal functions and an associated nonlinear interpolation operator, bridging approximation theory and multivariate fractal functions.

Contribution

It extends univariate fractal interpolation theory to multivariate functions, introducing a new class of multivariate $oldsymbol{ extalpha}$-fractal functions and a nonlinear interpolation operator.

Findings

Constructed a general framework for multivariate FIFs.

Introduced multivariate $oldsymbol{ extalpha}$-fractal functions as fractal approximants.

Studied properties of a multivariate fractal nonlinear interpolation operator.

Abstract

The natural kinship between classical theories of interpolation and approximation is well explored. In contrast to this, the interrelation between interpolation and approximation is subtle and this duality is relatively obscure in the context of fractal interpolation. The notion of $\alpha$-fractal function provides a proper foundation for the approximation theoretic facet of univariate fractal interpolation functions (FIFs). However, no comparable approximation theoretic aspects of FIFs has been developed for functions of several variables. The current article intends to open the door for intriguing interaction between approximation theory and multivariate FIFs. To this end, in the first part of this article, we develop a general framework to construct multivariate FIF, which is amenable to provide a multivariate analogue of the $\alpha$-fractal function. Multivariate $\alpha$-fractal functions provide a parameterized family of fractal approximants associated to a given multivariate continuous function. Some elementary aspects of the multivariate fractal nonlinear (not necessarily linear) interpolation operator that sends a continuous function defined on a hyper-rectangle to its fractal analogue is studied.