Exact solution of averaging procedure over the Cantor set

Using functional equations with self-similar properties, we have derived the exact analytical result for convolution of a smooth function with the normalized density of the Cantor set in the limit N→∞. We have proved that the self-similar kernel of this convolution cannot be reduced explicitly to any product of a power and a log-periodic function as suggested in literature. Only its asymptotic behaviour can be expressed in terms of such a product. This clarifies the relationship between fractals and fractional calculus.