Shift operator#Sequences

The shift operator {{mvar|T ^t}} (where {{tmath|t \in \R}}) takes a function {{mvar|f}} on {{tmath|\R}} to its translation {{mvar|f_t}},

: $T^t\; f(x)\; =\; f\_t(x)\; =\; f(x+t)~.$

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

{{Equation box 1

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which may be interpreted operationally through its formal Taylor expansion in {{mvar|t}}; and whose action on the monomial {{mvar|xⁿ}} is evident by the binomial theorem, and hence on all series in {{mvar|x}}, and so all functions {{math|f(x)}} as above.Jordan, Charles, (1939/1965). Calculus of Finite Differences, (AMS Chelsea Publishing), {{isbn|978-0828400336}} . This, then, is a formal encoding of the Taylor expansion in Heaviside's calculus.

The operator thus provides the prototypeM 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].

for Lie's celebrated advective flow for Abelian groups,

:$\backslash exp\backslash left(t\; \backslash beta(x)\; \backslash frac\{d\}\{dx\}\backslash right)\; f(x)\; =\; \backslash exp\backslash left(t\; \backslash frac\{d\}\{dh\}\backslash right)\; F(h)\; =\; F(h+t)\; =\; f\backslash left(h^\{-1\}(h(x)+t)\backslash right),$

where the canonical coordinates {{mvar|h}} (Abel functions) are defined such that

:$h\text{'}(x)\backslash equiv\; \backslash frac\; 1\; \{\backslash beta(x)\}\; ~,\; \backslash qquad\; f(x)\backslash equiv\; F(h(x)).$

For example, it easily follows that $\backslash beta\; (x)=x$ yields scaling,

:$\backslash exp\backslash left(t\; x\; \backslash frac\{d\}\{dx\}\backslash right)\; f(x)\; =\; f(e^t\; x)\; ,$

hence $\backslash exp\backslash left(i\backslash pi\; x\; \backslash tfrac\{d\}\{dx\}\backslash right)\; f(x)\; =\; f(-x)$ (parity); likewise,

$\backslash beta\; (x)=x^2$ yieldsp 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]

:$\backslash exp\backslash left(t\; x^2\; \backslash frac\{d\}\{dx\}\backslash right)\; f(x)\; =\; f\; \backslash left(\backslash frac\{x\}\{1-tx\}\backslash right),$

$\backslash beta\; (x)=\; \backslash tfrac\{1\}\{x\}$ yields

:$\backslash exp\backslash left(\backslash frac\{t\}\; \{x\}\; \backslash frac\{d\}\{dx\}\backslash right)\; f(x)\; =\; f\; \backslash left(\backslash sqrt\{x^2+2t\}\; \backslash right)\; ,$

$\backslash beta\; (x)=e^x$ yields

:$\backslash exp\backslash left\; (t\; e^x\; \backslash frac\; d\; \{dx\}\; \backslash right\; )\; f(x)\; =\; f\backslash left\; (\backslash ln\; \backslash left\; (\backslash frac\{1\}\{e^\{-x\}\; -\; t\}\; \backslash right\; )\; \backslash right\; )\; ,$

etc.

The initial condition of the flow and the group property completely determine the entire Lie flow, providing a solution to the translation functional equationAczel, J (2006), Lectures on Functional Equations and Their Applications (Dover Books on Mathematics, 2006), Ch. 6, {{isbn|978-0486445236}} .

:$f\_t(f\_\backslash tau\; (x))=f\_\{t+\backslash tau\}\; (x)\; .$

{{main|Shift space}}

The left shift operator acts on one-sided infinite sequence of numbers by

:$S^*:\; (a\_1,\; a\_2,\; a\_3,\; \backslash ldots)\; \backslash mapsto\; (a\_2,\; a\_3,\; a\_4,\; \backslash ldots)$

and on two-sided infinite sequences by

:$T:\; (a\_k)\_\{k\backslash ,=\backslash ,-\backslash infty\}^\backslash infty\; \backslash mapsto\; (a\_\{k+1\})\_\{k\backslash ,=\backslash ,-\backslash infty\}^\backslash infty.$

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

:$S:\; (a\_1,\; a\_2,\; a\_3,\; \backslash ldots)\; \backslash mapsto\; (0,\; a\_1,\; a\_2,\; \backslash ldots)$

and on two-sided infinite sequences by

:$T^\{-1\}:(a\_k)\_\{k\backslash ,=\backslash ,-\backslash infty\}^\backslash infty\; \backslash mapsto\; (a\_\{k-1\})\_\{k\backslash ,=\backslash ,-\backslash infty\}^\backslash infty.$

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

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 ^g}} maps {{math|F}} to"A one-parameter continuous group is equivalent to a group of translations". M Hamermesh, ibid.

:$F\_g(h)\; =\; F(h+g).$

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

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:

$$\backslash mathcal\{F\}\; T^t\; =\; M^t\; \backslash mathcal\{F\},$$

where {{mvar|M ^t}} 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 equal to all vectors which vanish in the first coordinate. The operator {{mvar|S}} is a compression of {{math|T{{i sup|−1}}}}, in the sense that

$$T^\{-1\}y\; =\; Sx\; \backslash text\{\; for\; each\; \}\; x\; \backslash in\; \backslash ell^2(\backslash N),$$

where {{mvar|y}} is the vector in {{tmath|\ell_2(\Z)}} with {{math|1=y_i = x_i}} for {{math|i ≥ 0}} and {{math|1=y_i = 0}} for {{math|i < 0}}. This observation is at the heart of the construction of many unitary dilations of isometries.

The spectrum of {{mvar|S}} is the unit disk. The shift {{mvar|S}} is one example of a Fredholm operator; it has Fredholm index −1.

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

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:

1. Associativity: let $(R^y\; f)(x)\; =\; (L^x\; f)(y).$ Then $L^x\; R^y\; =\; R^y\; L^x.$
2. There exists {{mvar|e}} in {{mvar|X}} such that {{mvar|L^e}} is the identity operator.

In this case, the set {{mvar|X}} is called a hypergroup.

• Arithmetic shift
• Logical shift
• Clock and shift matrices
• Finite difference
• Translation operator (quantum mechanics)

{{Reflist}}

• {{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.

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Category:Unitary operators