  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.
