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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.

There has been a considerable evolution of the theory of fractal interpolation function (FIF) over the last three decades. Recently, we introduced a multivariate analogue of a special class of FIFs, which is referred to as α-fractal functions, from the viewpoint of approximation theory. In the current note, we continue our study on multivariate α-fractal functions, but in the context of a few complete function spaces. For a class of fractal functions defined on a hyperrectangle Ω in the Euclidean space Rn, we derive conditions on the defining parameters so that the fractal functions are elements of some standard function spaces such as the Lebesgue spaces Lp(Ω), Sobolev spaces Wm,p(Ω), and Hölder spaces Cm,σ(Ω), which are Banach spaces. As a simple consequence, for some special choices of the parameters, we provide bounds for the Hausdorff dimension of the graph of the corresponding multivariate α-fractal function. We shall also hint at an associated notion of fractal operator that maps each multivariate function in one of these function spaces to its fractal counterpart. The latter part of this note establishes that the Riemann–Liouville fractional integral of a continuous multivariate α-fractal function is a fractal function of similar kind.

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