Runge type approximation results for spaces of smooth Whitney jets

We prove Runge type approximation results for linear partial differential operators with constant coefficients on spaces of smooth Whitney jets. Among others, we characterize when for a constant coefficient linear partial differential operator P(D) and for closed subsets F₁⊂ F₂ of Rᵈ the restrictions to F₁ of smooth Whitney jets f on F₂ satisfying $$P(D)f=0$$ on F₂ are dense in the space of smooth Whitney jets on F₁ satisfying the same partial differential equation on F₁ . For elliptic operators we give a geometric evaluation of this characterization. Additionally, for differential operators with a single characteristic direction, like parabolic operators, we give a sufficient geometric condition for the above density to hold. Under mild additional assumptions on ∂ F₁ and for F₂=Rᵈ this sufficient conditions is also necessary. As an application of our work, we characterize those open subsets Ω of the complex plane satisfying Ω =intΩ ̄ for which the set of holomorphic polynomials are dense in A^∞ (Ω ) , under the additional hypothesis that Ω ̄ satisfies the strong regularity condition. Furthermore, for the wave operator in one spatial variable, a simple sufficient geometric condition on F₁, F₂⊂ R² is given for the above density to hold. For the special case of F₂=R² this sufficient condition is also necessary under mild additional hypotheses on F₁ .