Finite Normal Forms and Causal Inversion for Fractional Operator Algebras

Abstract

Fractional operator algebras provide a natural setting for linking algebraic normal forms to causal inversion, but arbitrary smooth multipliers and Volterra kernels destroy finite fractional rewriting. We construct an algebra generated by polynomial multipliers, fractional powers indexed by an additive subgroup $Γ\subset\mathbb{R}$ containing $\mathbb{Z}$, and flat causal convolution operators. Its finite polynomially weighted diagonal-flat convolution operators form the smallest two-sided ideal containing the flat generators, polynomial multipliers are intrinsic, and distributional separation yields a unique finite normal form with an additive order filtration. For monic normal forms, the leading power selects a canonical causal right factor that extends from the flat core to the natural range $J^αC[0,T]$. Completing the negative-order convolution sector in $L^1$ admits integrable Sonine and Prabhakar kernels, while an independently prescribed continuous bivariate Volterra term remains external. Under injectivity and power-type kernel bounds, variable lower-order terms and this bivariate perturbation reduce to one weakly singular second-kind equation, whose classical resolvent exists on every finite interval without smallness assumptions. A finite boundary-jet block incorporates prescribed initial data without changing the reduced kernel. Sonine, distributed-order, and regularized-Prabhakar realizations follow. In the noninteger Prabhakar regime, an exact range criterion characterizes the classical $AC^m$ subdomain, and an explicit Riemann--Liouville example proves that it is strictly smaller than the continuous natural range.