Approximation properties of bivariate α-fractal functions and dimension results

Non-Stationary $$\alpha $$-Fractal Surfaces

“…They also expressed the fractal dimension in terms of different classical oscillation measures and in terms of wavelet expansions by comparing the oscillation spaces to certain Besov spaces. In [14,28,29], box dimensions of (stationary) α-fractal functions are estimated using Hölder spaces and variation method, however, the case of oscillation spaces are not discussed. Our result may generalize the results available in [15,17], because bilinear FIS is a particular case of (stationary) α-fractal function, see, for instance, [30,Remark 2.1].…”

“…Motivated by the seminal work [1] of Barnsley on Fractal Interpolation Functions (FIFs), Navascués [23] introduced a family of fractal interpolation functions f α known as α-fractal functions corresponding to a continuous function f on closed and bounded interval of R. The α−fractal function f α approximates and interpolates f simultaneously. As it is evident from some works of Navascués and her collaborators [8,9,14,[22][23][24]28,30,32] that there are enormous applications of (stationary) fractal functions in different areas of mathematics.…”

“…In [29,30], Verma and Viswanathan introduced and studied the bivariate α-fractal functions from the fractal geometry, operator theoretic and constrained approximation point of view. In [14], Jha et al obtained some bounds of the box dimension of α-fractal function. They also focus on some approximation and smoothness properties of bivariate α-fractal function.…”

On Bivariate Fractal Interpolation for Countable Data and Associated Nonlinear Fractal Operator

“…Among various constructions available in the literature, we found the general framework to construct fractal surfaces given in [25] to be quite interesting. This is due to the fact that the construction thereat is amenable to obtain bivariate analogue of α-fractal function, which is a natural entry point to delve into the theory of bivariate fractal approximation, see also [12,28].…”