A result of Meise and Taylor in 1988 shows that every non-zero convolution operator on the Frechet space $H(\mathbb{C})$ of all entire functions on $\mathbb{C}$ has a continuous linear right inverse $R$. Upon skimming that paper, it seems that no explicit formulation of $R$ is given - the existence of $R$ is implied by some algebraic results in the paper.
Each convolution operator can be represented as a function $\varphi(D)$ of the derivative operator $D$, where $\varphi(z)$ is an entire function for which there exist positive constants $a,b$ such that $|\varphi(z)|\leq ae^{b|z|}$. In a simple case like $\varphi(D)=I+D$, where $I$ is the identity operator, is there an explicit formulation for its continuous linear right inverse $R$? What I want to do is study $R$ in a basic way, by say computing the image of functions like $z^2$ or $e^z$ under $R$.
