Box Dimension and Fractional Integrals of Multivariate Fractal Interpolation Functions

Abstract

In this article, we construct the multivariate fractal interpolation functions for a given data points and explore the existence of $\alpha$-fractal function corresponding to the multivariate continuous function defined on $[0,1]\times \cdots \times [0,1](q\text{-times})$. The parameters are selected such that the corresponding fractal version preserves some of the original function's properties, for instance, if the given function is H\"older continuous, then the corresponding $\alpha$-fractal function is also H\"older continuous. Moreover, we explore the restriction of the $\alpha$-fractal function on the co-ordinate axis. Furthermore, the box dimension and Hausdorff dimension of the graph of the multivariate $\alpha$-fractal function and its restriction are investigated. In the last section, we prove that the mixed Riemann-Liouville fractional integral of fractal function satisfies a self-referential equation.