Box Dimension and Fractional Integrals of Multivariate Fractal Interpolation Functions

TL;DR

This paper constructs multivariate fractal interpolation functions, explores their properties, dimensions, and fractional integrals, extending fractal analysis to higher dimensions with preserved properties.

Contribution

It introduces a method to create multivariate fractal functions that preserve properties like Hölder continuity and investigates their dimensions and fractional integrals.

Findings

Fractal functions preserve Hölder continuity.

Dimensions of fractal functions are analyzed.

Fractional integrals satisfy self-referential equations.

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.