On Approximation Properties of Fractional Integral for A-Fractal Function

In this paper, the Riemann–Liouville fractional integral of an A-fractal function is explored by taking its vertical scaling factors in the block matrix as continuous functions from 0,1 to ℠. As the scaling factors play a significant role in the generation of fractal functions, the necessary condition for the scaling factors in the block matrix is outlined for the newly obtained function. The resultant function of the fractional integral is demonstrated as an A-fractal function if the scaling factors obey the necessary conditions. Furthermore, this article proposes a fractional operator which defines the Riemann–Liouville fractional integral of an A-fractal function for each continuous function on CI,℠2, where CI,℠2 is the space of all continuous functions from closed interval I⊂℠to ℠2. In addition, the approximation properties such as linearity, boundedness, and semigroup property of the proposed fractional operator are investigated.