Eur. Phys. J. Spec. Top. (2021) 230: 3807-3814
https://doi.org/10.1140/epjs/s11734-021-00327-2

Regular Article

Fractal dimension of Katugampola fractional integral of vector-valued functions

¹ Department of Mathematical Sciences, Indian Institute of Technology (Banaras Hindu University), 221005, Varanasi, India
² Department of Applied Sciences, Indian Institute of Information Technology Allahabad, 211015, Prayagraj, India

Received: 28 August 2021
Accepted: 30 October 2021
Published online: 30 November 2021

Abstract

Calculating fractal dimension of the graph of a function not simple even for real-valued functions. While through this paper, our intention is to provide some initial theories for the dimension of the graphs of vector-valued functions. In particular, we give a fresh attempt to estimate the fractal dimension of the graph of the Katugampola fractional integral of a vector-valued continuous function of bounded variation defined on a closed bounded interval in $\mathbb{R}.$ We prove that dimension of the graph of a continuous vector-valued function of bounded variation is 1 and so is the dimension of the graph of its Katugampola fractional integral. Further, for a Hölder continuous function, we provide an upper bound for the upper box dimension of the graph of each coordinate function of the Katugampola fractional integral of the function.