Shift operator

Summary

A shift operator moves the argument of a function or the index of a sequence by a fixed amount. On functions of a real variable it sends $f$ to $x \mapsto f(x + h)$; on sequences it sends $(u_n)$ to $(u_{n+1})$. It is the building block of the operator calculus of finite differences, since the forward difference is the shift minus the identity. In functional analysis the shift on sequence spaces is a basic example of a linear operator.