Editing Shift operator

{{short description|Linear mathematical operator which translates a function}}
{{About|shift operators in mathematics|operators in computer programming languages|Bit shift|the shift operator of group schemes|Verschiebung operator}}

In [[mathematics]], and in particular [[functional analysis]], the '''shift operator''', also known as the '''translation operator''', is an [[Operator (mathematics)|operator]] that takes a [[Function (mathematics)|function]] {{math|''x'' ↦ ''f''(''x'')}}
to its '''translation''' {{math|''x'' ↦ ''f''(''x'' + ''a'')}}.<ref>{{MathWorld|id=ShiftOperator|title=Shift Operator}}</ref> In [[time series analysis]], the shift operator is called the ''[[lag operator]]''.

Shift operators are examples of [[linear operator]]s, important for their simplicity and natural occurrence. The shift operator action on [[Function of a real variable|functions of a real variable]] plays an important role in [[harmonic analysis]], for example, it appears in the definitions of [[almost periodic function#Uniform or Bohr or Bochner almost periodic functions|almost periodic functions]], [[positive-definite function]]s, [[derivative]]s, and [[convolution]].<ref name=mar>{{cite book|mr=2182783|last=Marchenko|first=V. A.|author-link=Vladimir Marchenko|chapter=The generalized shift, transformation operators, and inverse problems|title=Mathematical events of the twentieth century|pages=145&ndash;162|publisher=Springer|location=Berlin|year=2006|doi=10.1007/3-540-29462-7_8|isbn=978-3-540-23235-3 }}</ref> Shifts of sequences (functions of an integer variable) appear in diverse areas such as [[Hardy space]]s, the theory of [[abelian variety|abelian varieties]], and the theory of [[symbolic dynamics]], for which the [[baker's map]] is an explicit representation. The notion of [[triangulated category]] is a [[categorification | categorified]] analogue of the shift operator.

==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>

==Properties of the shift operator==

The shift operator acting on real- or complex-valued functions or sequences is a linear operator  which preserves most of the standard [[norm (mathematics)|norms]] which appear in functional analysis. Therefore, it is usually a [[continuous operator]] with norm one.

===Action on Hilbert spaces===

The shift operator acting on two-sided sequences is a [[unitary operator]] on  {{tmath|\ell_2(\Z).}} The shift operator acting on functions of a real variable is a unitary operator on {{tmath|L_2(\R).}}

In both cases, the (left) shift operator satisfies the following commutation relation with the Fourier transform:
<math display="block"> \mathcal{F} T^t = M^t \mathcal{F}, </math>
where {{mvar|M<sup> t</sup>}} is the [[multiplication operator]] by {{math|exp(''itx'')}}. Therefore, the spectrum of {{mvar|T{{sup| t}}}} is the unit circle.

The one-sided shift {{mvar|S}} acting on {{tmath|\ell_2(\N)}} is a proper [[isometry]] with [[range of a function|range]] equal to all [[Vector (geometric)|vectors]] which vanish in the first [[coordinate]]. The operator {{mvar|S}} is a [[compression (functional analysis)|compression]] of {{math|''T''{{i sup|−1}}}}, in the sense that
<math display="block">T^{-1}y = Sx \text{ for each } x \in \ell^2(\N),</math>
where {{mvar|y}} is the vector in {{tmath|\ell_2(\Z)}} with {{math|1=''y<sub>i</sub>'' = ''x<sub>i</sub>''}} for {{math|''i'' &ge; 0}} and {{math|1=''y<sub>i</sub>'' = 0}} for {{math|''i'' < 0}}. This observation is at the heart of the construction of many [[unitary dilation]]s of isometries.

The [[Spectrum (functional analysis)|spectrum]] of {{mvar|S}} is the [[unit disk]]. The shift {{mvar|S}} is one example of a [[Fredholm operator]]; it has Fredholm index&nbsp;−1.

==Generalization==

[[Jean Delsarte]] introduced the notion of '''generalized shift operator''' (also called '''generalized displacement operator'''); it was further developed by [[Boris Levitan]].<ref name = mar/><ref>{{SpringerEOM|id=g/g043800|first=B.M.|last=Levitan|author-link=Boris Levitan|first2=G.L.|last2=Litvinov|title=Generalized displacement operators}}</ref><ref>{{SpringerEOM|id=A/a011970|first=E.A.|last= Bredikhina|title=Almost-periodic function}}</ref>

A family of operators {{tmath|\{L^x\}_{x \in X} }} acting on a space {{math|Φ}} of functions from a set {{mvar|X}} to {{tmath|\C}} is called a family of generalized shift operators if the following properties hold:
# [[Associative property|Associativity]]: let <math>(R^y f)(x) = (L^x f)(y).</math> Then <math>L^x R^y = R^y L^x.</math>
# There exists {{mvar|e}} in {{mvar|X}} such that {{mvar|L<sup>e</sup>}} is the [[Identity function|identity operator]].
In this case, the set {{mvar|X}} is called a [[hypergroup]].

==See also==
*[[Arithmetic shift]]
*[[Logical shift]]
*[[Clock and shift matrices]]
*[[Finite difference#Calculus of finite differences|Finite difference]]
*[[Translation operator (quantum mechanics)]]

==Notes==
{{Reflist}}

==Bibliography==
* {{cite book | last=Partington | first=Jonathan R. | title=Linear Operators and Linear Systems | publisher=Cambridge University Press | date=March 15, 2004 | isbn=978-0-521-83734-7 | doi=10.1017/cbo9780511616693}}
* Marvin Rosenblum and James Rovnyak, ''Hardy Classes and Operator Theory'', (1985) Oxford University Press.

{{DEFAULTSORT:Shift Operator}}
[[Category:Unitary operators]]