Fractal dimension of fractional integral of continuous functions

Abstract

This paper mainly makes research on fractal dimension of fractal functions. We give basic estimation of fractal dimension, such as Box dimension and Hausdorff dimension, of fractional calculus of continuous functions. For a continuous function, upper Box dimension of its Riemann-Liouville fractional integral of order v has been proved to be no more than 2 − v when 0 ≤ v ≤ 1. Furthermore, if a continuous function which satisfies α-Holder condition, upper Box dimension of its Riemann-Liouville fractional integral is no more than 2 − α when 0 ≤ α ≤ 1. This means upper Box dimension of Riemann-Liouville fractional integral of a continuous function satisfying α-Holder condition of order v is no more than min{2 − v, 2 − α} when 0 ≤ v, α ≤ 1. With method of auxiliary function, upper Box dimension of Riemann-Liouville fractional integral of any continuous functions satisfying α-Holder condition of order v is strictly less than min{2 − v, 2 − α} when 0 ≤ v, α ≤ 1.