Fractal dimension of Katugampola fractional integral of vector-valued functions

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 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. © 2021, The Author(s), under exclusive licence to EDP Sciences, Springer-Verlag GmbH Germany, part of Springer Nature.