Translation finite-difference operator

On all real-valued functions on $\mathbb{R}$, define $D_{r}f(x)=f(x+r)-f(x)$. The operators commute, and every real number except $-1$ is an eigenvalue. For eigenvalue $\lambda \ne -1$, values may be chosen independently on the cosets of $r\mathbb{Z}$ and extended by $f(x+r)=(1+\lambda )f(x)$, so every eigenspace is infinite-dimensional.

If $p$ has degree $n$ and leading coefficient $a_n$, then for nonzero steps $r_1,\dots ,r_n$,Consequently a degree-$n$ polynomial cannot be the sum of $n$ functions each periodic with some nonzero period.