Box Dimension and Fractional Integrals of Multivariate α -Fractal Functions

Abstract

In this article, we construct multivariate fractal interpolation functions for a given set of data points and explore the existence of the α-fractal function corresponding to the multivariate continuous function defined on [0 , 1] × ⋯ × [0 , 1] (q-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 Hölder continuous, then the corresponding α-fractal function is also Hölder continuous. Moreover, we explore the restriction of the α-fractal function on the coordinate axis. Furthermore, the box dimension and Hausdorff dimension of the graph of the multivariate α-fractal function and its restriction are investigated. Later, we prove that the mixed Riemann–Liouville fractional integral of a fractal function satisfies a self-referential equation. © 2023, The Author(s), under exclusive licence to Springer Nature Switzerland AG.