Editing Shift operator (section)

==Definition==

===Functions of a real variable===

The shift operator {{mvar|T<sup> t</sup>}} (where {{tmath|t \in \R}}) takes a function {{mvar|f}} on {{tmath|\R}} to its translation {{mvar|f<sub>t</sub>}}, 
: <math>T^t f(x) =  f_t(x) = f(x+t)~.</math>

A practical [[operational calculus]] representation of the linear operator {{mvar|T<sup> t</sup>}} in terms of the plain derivative {{tmath|\tfrac{d}{dx} }} was introduced by [[Lagrange]],

{{Equation box 1
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|equation = <math>T^t= e^{t \frac d {dx}}~, </math>
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which may be interpreted operationally through its formal [[Taylor expansion]] in {{mvar|t}}; and whose action on the monomial {{mvar|x<sup>n</sup>}} is evident by the [[binomial theorem]], and hence  on ''all series in'' {{mvar|x}}, and so all functions {{math|''f''(''x'')}} as above.<ref>Jordan, Charles, (1939/1965). ''Calculus of Finite Differences'', (AMS Chelsea Publishing), {{isbn|978-0828400336}} .</ref> This, then, is a formal encoding of the Taylor expansion in Heaviside's calculus.

The operator thus provides the prototype<ref>M Hamermesh (1989), ''Group Theory and Its Application to Physical Problems'' 
(Dover Books on Physics), Hamermesh ISBM 978-0486661810, Ch 8-6, pp 294-5, 
[https://physics.stackexchange.com/questions/331635/undefined-phase-flow/331841#331841 online].</ref> 
for Lie's celebrated [[Iterated function#Lie's data transport equation|advective flow for Abelian groups]],
:<math> \exp\left(t \beta(x) \frac{d}{dx}\right) f(x) = \exp\left(t \frac{d}{dh}\right) F(h) = F(h+t) = f\left(h^{-1}(h(x)+t)\right),</math>
where the canonical coordinates {{mvar|h}} ([[Abel equation|Abel functions]]) are defined such that 
:<math>h'(x)\equiv \frac 1 {\beta(x)} ~, \qquad f(x)\equiv F(h(x)). </math>

For example, it easily follows that  <math>\beta (x)=x</math> yields scaling,
:<math> \exp\left(t x \frac{d}{dx}\right) f(x) =   f(e^t x) , </math>
hence <math> \exp\left(i\pi x \tfrac{d}{dx}\right) f(x) = f(-x)</math> (parity); likewise, 
<math>\beta (x)=x^2</math> yields<ref>p 75 of  Georg Scheffers (1891): ''Sophus Lie, Vorlesungen Ueber Differentialgleichungen Mit Bekannten Infinitesimalen Transformationen'', Teubner, Leipzig, 1891. {{isbn|978-3743343078}}  [https://books.google.com/books?id=7-86AQAAIAAJ&q=+75&pg=PR6 online]
</ref> 
:<math> \exp\left(t x^2 \frac{d}{dx}\right) f(x) = f \left(\frac{x}{1-tx}\right),</math>

<math>\beta (x)= \tfrac{1}{x}</math> yields 
:<math> \exp\left(\frac{t} {x} \frac{d}{dx}\right) f(x) = f \left(\sqrt{x^2+2t} \right) ,</math>

<math>\beta (x)=e^x</math> yields 
:<math> \exp\left (t e^x \frac d {dx} \right ) f(x) = f\left (\ln \left (\frac{1}{e^{-x} - t} \right ) \right )  ,</math>
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<ref  name="acz">Aczel, J (2006), ''Lectures on Functional Equations and Their Applications'' (Dover Books on Mathematics, 2006), Ch. 6, {{isbn|978-0486445236}} .</ref>
:<math>f_t(f_\tau (x))=f_{t+\tau} (x)  .</math>

===Sequences===
{{main|Shift space}}
The '''left shift''' operator acts on one-sided [[infinite sequence]] of numbers by

:<math> S^*: (a_1, a_2, a_3, \ldots) \mapsto (a_2, a_3, a_4, \ldots)</math>

and on two-sided infinite sequences by

:<math> T: (a_k)_{k\,=\,-\infty}^\infty \mapsto (a_{k+1})_{k\,=\,-\infty}^\infty.</math>

The '''right shift''' operator acts on one-sided [[infinite sequence]] of numbers by

:<math> S: (a_1, a_2, a_3, \ldots) \mapsto (0, a_1, a_2, \ldots)</math>

and on two-sided infinite sequences by

:<math> T^{-1}:(a_k)_{k\,=\,-\infty}^\infty \mapsto (a_{k-1})_{k\,=\,-\infty}^\infty.</math>

The right and left shift operators acting on two-sided infinite sequences are called '''''bilateral''''' shifts.

===Abelian groups===

In general, as illustrated above, if {{mvar|F}} is a function on an [[abelian group]] {{mvar|G}}, and {{mvar|h}} is an element of {{mvar|G}}, the shift operator {{mvar|T<sup> g</sup>}} maps {{math|''F''}} to<ref name="acz" /><ref>"A one-parameter continuous group is equivalent to a group of translations". M Hamermesh, ''ibid''.</ref>
:<math> F_g(h) = F(h+g).</math>