RELATIONSHIP OF UPPER BOX DIMENSION BETWEEN CONTINUOUS FRACTAL FUNCTIONS AND THEIR RIEMANN–LIOUVILLE FRACTIONAL INTEGRAL

Abstract

This paper considers the relationship of box dimension between a continuous fractal function and its Riemann–Liouville fractional integral. For an arbitrary fractal function [Formula: see text] it is proved that the upper box dimension of the graph of Riemann–Liouville fractional integral [Formula: see text] does not exceed the upper box dimension of [Formula: see text], i.e. [Formula: see text]. This estimate shows that [Formula: see text]order Riemann–Liouville fractional integral [Formula: see text] does not increase the fractal dimension of the integrand [Formula: see text], which means that Riemann–Liouville fractional integration does not decrease the smoothness at least that is obvious known result for classic integration. Our result partly answers fractal calculus conjecture in [F. B. Tatom, The relationship between fractional calculus and fractals, Fractals 2 (1995) 217–229] and [Y. S. Liang and W. Y. Su, Riemann–Liouville fractional calculus of one-dimensional continuous functions, Sci. Sin. Math. 4 (2016) 423–438].