The relationship between the fractional integral and the fractal structure of a memory set

Abstract

It is shown that there is no direct relation between the fractional exponent ν of the fractional integral and the fractal structure of the memory set considered, ν depends only on the first contraction coefficient ξ1 and the first weight P1 of the self-similar measure (or infinite self-similar measure) μ on the memory set. If and only if P1 = ξβ 1 (where β ∈ (0, 1) is the fractal dimension of the memory set), ν is equal to the fractal dimension of the memory set. It is also true that ν is continuous about ξ1 and P1.