We give two kinds of direct sum decompositions of the class of infinitely differentiabe functions. They are related to the kernel of the higher order partial derivative operator and the polynomial space.
"Partial primitives, polyprimitives and decompositions of the class of infinitely differentiable functions." Hiroshima Math. J. 38 (2) 203 - 223, July 2008. https://doi.org/10.32917/hmj/1220619457