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 of the complex plane satisfying = int for which the set of holomorphic polynomials are dense in A^ (), under the additional hypothesis that 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.
Ciaś et al. (Thu,) studied this question.