Partially Explore the Differences and Similarities between Riemann-Liouville Integral and Mellin Transform

Abstract

At present many researchers devote themselves to studying the relationship between continuous fractal functions and their fractional integral. But little attention is paid to the relationship between Mellin transform and fractional integral. This paper aims to partially explore the differences and similarities between Riemann-Liouville integral and Mellin transform, then a 1-dimensional continuous and unbounded variational function defined on the closed interval [0,1] needs to be constructed. Through describing the image of the constructed function and its transformed function and proving the relevant properties, we obtain that Box dimension of its Riemann–Liouville integral of arbitrary order and its Mellin transformed function are also one. The smoothness of its Riemann–Liouville integral can only be improved, and its Mellin transformed function is differentiable.