A Spectral Algebra for Multidimensional Weyl Fractional Operators on Exponential Characters

Let d. We show that multidimensional Weyl fractional operators admit a natural spectral realization on the algebraic span of exponential characters. More precisely, if (C_+) ᵈ and e_ (x): =e^, x, , \ xᵈ, where C_+: =\z: z>0\, then the algebraic direct sum E_^alg: = _Ce_carries a canonical diagonal action of the generalized multidimensional Weyl integrals and derivatives. If Iₖ, ₀=I₁^a₁ Id^ad Dₖ, ₀, ₌=D₁^a₁, m₁ Dd^ad, mdare the coordinatewise generalized Weyl integral and derivative determined by a kernel tuple a= (a₁, , ad) and an integer multi-index m₀ᵈ, then Iₖ, ₀e_=a () e_ Dₖ, ₀, ₌e_=₀, ₌ () e_, with symbols a (): =₉=₁ᵈ aⱼ (ⱼ), ₀, ₌ (): =₉=₁ᵈ ⱼ^mⱼaⱼ (ⱼ). Thus the corresponding operator algebra is diagonal and isomorphic to the algebra of scalar multipliers on. We further prove the law of exponents by symbol multiplication: Iₖ, ₀Iₖ, ₁=Iₖ, ₀*䃐 ₁, Dₖ, ₀, ₌Dₖ, ₁, ₍=Dₖ, ₀*䃐 ₁, ₌+₍. For the standard Weyl fractional derivative W^{} of order [0, ) ᵈ, we obtain W^{}e_=^{}e_, ^{}: =₉=₁ᵈ ⱼ^ⱼ, so the Weyl calculus becomes a genuine spectral algebra on E_^alg. This stands in contrast to the shift-algebra models for Riemann-Liouville and Caputo operators on half-spaces: on Rᵈ there is no vacuum defect and no boundary layer.