Fractional Calculus - A Commutative Method on Real Analytic Functions

The traditional first approach to fractional calculus is via the Riemann-Liouville differintegral ${}_a{D}_x^k$. The intent of this paper will be to create a space $K$, pair of maps $g:C^{\omega }(\mathbb{R})\to K$ and $g^{\mathrm{\prime }}:K\to C^{\omega }(\mathbb{R}$), and operator $D^k:K\to K$ such that the operator $D^k$ commutes with itself, the map $g$ embeds $C^{\omega }(\mathbb{R}$) isomorphically into $K$, and the following diagram commutes; \xymatrix{C^{\omega}($\mathbb{R}$) \ar[d]{{a}D_{x}^{k}} \ar[r]^{g} & K \ar[d]^{D^{k}} C^{\omega}($\mathbb{R}$) & K \ar[l]^{g’}} \qquad This implies the following diagram commutes, for analytic $f$ such that ${}_a{D}_x^{j}f$ = 0 (i.e, if $f={\sum }_{i\in I}{b}_i$(x-$a{)}^i$, where {b_{i}} \subset \mathbb{R}, and $I \subseteq {j-1, …, j-\lfloor j \rfloor$}); \xymatrix{f \ar@/3pc/[dd]{{a}D{x}^{j+k}} \ar[d]^{{a}D{x}^{j}} \ar[r]^{g} & g(f) \ar[d]^{D^{j}} 0 & \ar[l]^{g’} D^{j}g(f) \ar[d]^{D^{k}} {a}D{x}^{j+k}f &\ar[l]^{g’} D^{k}D^{j}g(f)}