In traditional interpolation theories, the relationship between interpolation and approximation is extensively studied. However, in the realm of fractal interpolation, this interplay remains subtle and relatively unexplored, especially in several variables. The concept of α-fractal functions establishes a foundation for understanding the approximation aspect of univariate fractal interpolation functions (FIFs). However, the extension of these approximation principles to functions of several variables remains unaddressed.

In this talk, we shall explore some of these interplay between interpolation and approximation of multivariate functions in the fractal setting, drawing from joint work with Viswanathan [Electron. Trans. Numer. Anal. 55 (2022), 627–651]. We will present a general framework for constructing multivariate FIFs, laying the groundwork for a multivariate analogue of α-fractal functions. These functions offer a parameterized family of fractal approximants for mul-
tivariate continuous functions. We will delve into fundamental aspects of the multivariate fractal interpolation operator, which takes a continuous function defined on a hyperrectangle into its fractal counterpart. Analogous to the univariate scenario, α-fractal functions form the basis for extending fractalization to various results in multivariate approximation theory.