Translation finite-difference operator
= Translation finite-difference operator
On all real-valued functions on $\mathbb R$, define $D_rf(x)=f(x+r)-f(x)$. 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.