Abelian Groups of Fractional Operators

Proposal for Use of the Fractional Derivative of Radial Functions in Interpolation Problems

“…Fractional operators have various representations, but one of their fundamental properties is that they recover the results of conventional calculus when α → n. Before continuing, it is worth mentioning that due to the large number of fractional operators that exist [38][39][40][41][42][43][44][45][46][47][48][49][50][51][52][53][54][55][56], it seems that the most natural way to fully characterize the elements of fractional calculus is by using sets, which is the main idea behind the methodology known as fractional calculus of sets [57][58][59][60], whose seed of origin is the fractional Newton-Raphson method [24]. Therefore, considering a scalar function h : R m → R and the canonical basis of R m denoted by { êk } k≥1 , it is feasible to define the following fractional operator of order α using Einstein's notation:…”

“…Before proceeding, it is worth mentioning that some applications may be derived based on the previous definition, among which the following corollary can be found [59,60]:…”

“…On the other hand, it is important to highlight that Corollary 1 allows generating groups of fractional operators under other operations [59]. For example, considering the following operation…”