Approximation of bivariate Lipschitz continuous functions by hidden variable fractal functions

The crux of the present paper is approximation of bivariate Lipschitz continuous functions by hidden variable fractal functions. We propose the construction of hidden variable fractal perturbation associated with a given bivariate Lipschitz continuous function defined on a rectangle \(\mathcal {D}\) in the Euclidean space. This procedure yields a fractal operator on the space of all \(\mathbb {R}^2\)-valued Lipschitz continuous functions defined on a rectangle \(\mathcal {D}\). Some basic and important properties of this fractal operator will be discussed. Subsequently, we extend this fractal operator to the norm preserving bounded linear operator on the the space of all \(\mathbb {R}^2\)-valued continuous functions defined on a rectangle \(\mathcal {D}\). We investigate the stability of hidden variable fractal functions with respect to a perturbation in the scaling factors. Finally, existence of optimal hidden variable fractal function which approximates the given bivariate Lipschitz continuous function is discussed.