Functions of a real variable
The shift operator T t (where ) takes a function f on to its translation ft,
A practical operational calculus representation of the linear operator T t in terms of the plain derivative was introduced by Lagrange,
which may be interpreted operationally through its formal Taylor expansion in t; and whose action on the monomial xn is evident by the binomial theorem, and hence on all series in x, and so all functions f(x) as above.[3] This, then, is a formal encoding of the Taylor expansion in Heaviside's calculus.
The operator thus provides the prototype[4]
for Lie's celebrated advective flow for Abelian groups,
where the canonical coordinates h (Abel functions) are defined such that
For example, it easily follows that yields scaling,
hence (parity); likewise,
yields[5]
yields
yields
etc.
The initial condition of the flow and the group property completely determine the entire Lie flow, providing a solution to the translation functional equation[6]
Sequences
The left shift operator acts on one-sided infinite sequence of numbers by
and on two-sided infinite sequences by
The right shift operator acts on one-sided infinite sequence of numbers by
and on two-sided infinite sequences by
The right and left shift operators acting on two-sided infinite sequences are called bilateral shifts.