An approximation theoretic revamping of fractal interpolation surfaces on triangular domains

The theory of fractal surfaces, in its basic setting, asserts the existence of a bivariate continuous function defined on a triangular domain. The extant literature on the construction of fractal surfaces over triangular domains use certain assumptions for the construction and deal primarily with the interpolation aspects. Working in the framework of fractal surfaces over triangular domains, this note has a two-fold target. Firstly, to revamp the existing constructions of fractal surfaces over triangular domains, and secondly to connect the idea of fractal surfaces further with the theory of approximation. To this end, in the same spirit of the so-called \(\alpha \)-fractal functions on intervals and hyperrectangles, we study a salient subclass of fractal surfaces, which provides a parameterized family of bivariate fractal functions corresponding to a fixed continuous function defined on a triangular domain. Some elementary properties of the single-valued (linear and nonlinear) and multi-valued fractal operators associated with the \(\alpha \)-fractal function formalism of the bivariate fractal functions on a triangular domain are recorded. A fractal approximation process, that is a sequence of single-valued fractal operators converging strongly to the identity operator on \({\mathcal {C}}(\Delta , {\mathbb {R}})\), the space of all real-valued continuous functions defined on a triangular domain \(\Delta \), is obtained. An approximation class of fractal functions, referred to as the fractal polynomials, is hinted at. The notion of \(\alpha \)-fractal function and associated single-valued fractal operator in conjunction with appropriate stability results for Schauder bases provide Schauder bases consisting of self-referential functions for \({\mathcal {C}}(\Delta , {\mathbb {R}})\).