On the averaging procedure over the Cantor set

Abstract

The procedure of averaging a smooth function over the normalized density of the Cantor set (A. Le Mehaute, R.R. Nigmatullin, L. Nivanen. Fleches du temps et geometric fractale. Paris: “Hermes”, 1998, Chapter 5) has been shown not to reduce exactly the convolution to the classical fractional integral of Riemann-Liouville type. Although the asymptotic behavior of the self-similar convolution kernel is very close to the product of a power and a log-periodic function, this is not obviously enough to claim the direct relationship between the fractals and the fractional calculus.