Fractional Calculus of Fractal Functions on Weighted Sobolev Spaces

In this article, the α-fractal interpolation function fα corresponding to any function f belonging to the weighted Sobolev space Wρr,2(I) is defined. The convergence of sequences of α-fractal interpolation functions corresponding to mappings in Wρr,2(I) with respect to the uniform norm as well as the weighted Sobolev norm is discussed. It is proved that the Riemann–Liouville fractional order integral of an α-fractal interpolation function of any map f∈Wρr,2(I) is also a self-referential function interpolating a specific data set. Some aspects of the convergence of the Riemann–Liouville integral of α-fractal functions when the original mappings converge are also analyzed. In short, by imposing certain conditions on the base function and the scale vector of a specific iterated function system, fractal perturbations of functions from weighted Sobolev spaces are defined. It is also proved that, under suitable hypotheses, the Riemann–Liouville fractional integral of these fractal mappings on Sobolev spaces is a fractal function of the same kind.