Shift operator

In mathematics, and in particular functional analysis, the shift operator also known as translation operator is an operator that takes a function $x \mapsto f (x)$ to its translation $x \mapsto f (x + a)$ .^[1] In time series analysis, the shift operator is called the lag operator.

Shift operators are examples of linear operators, important for their simplicity and natural occurrence. The shift operator action on functions of a real variable plays an important role in harmonic analysis, for example, it appears in the definitions of almost periodic functions, positive definite functions, and convolution.^[2] Shifts of sequences (functions of an integer variable) appear in diverse areas such as Hardy spaces, the theory of abelian varieties, and the theory of symbolic dynamics, for which the baker's map is an explicit representation.

Definition

Functions of a real variable

The shift operator $T t$ ( $t \in R$ ) takes a function $f$ on R to its translation $f t$ ,

T^{t}f(x)=f_{t}(x)=f(x+t)~.

A practical representation of the linear operator $T t$ in terms of the plain derivative $d ⁄ dx$ was introduced by Lagrange,

T^{t}=e^{t{\frac {d}{dx}}}~,

which may be interpreted operationally through its formal Taylor expansion in $t$ ; and whose action on the monomial $x n$ 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.

The operator thus provides the prototype^[4] for Lie's celebrated advective flow for Abelian groups,

e^{t\beta (x){\frac {d}{dx}}}f(x)=e^{t{\frac {d}{dh}}}F(h)=F(h+t)=f(h^{-1}(h(x)+t)),

where the canonical coordinates $h$ (Abel functions) are defined, s.t.

h'(x)\equiv {\frac {1}{\beta (x)}}~,\qquad f(x)\equiv F(h(x)).

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

e^{tx{\frac {d}{dx}}}f(x)=f(e^{t}x)

and $\beta (x)=x^{2}$ yields^[5]

e^{tx^{2}{\frac {d}{dx}}}f(x)=f\left({\frac {x}{1-tx}}\right)

$\beta (x)=1/x$ yields

e^{{\frac {t}{x}}{\frac {d}{dx}}}f(x)=f({\sqrt {x^{2}+2tx}}~)

$\beta (x)=e^{x}$ yields

\exp \left(e^{x}{\frac {d}{dx}}\right)f(x)=f\left(\ln \left({\frac {1}{\exp(-x)-t}}\right)\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 equation^[6]

f_{t}(f_{\tau }(x))=f_{t+\tau }(x).

Sequences

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

S^*: (a_1, a_2, a_3, \ldots) \mapsto (a_2, a_3, a_4, \ldots)

and on two-sided infinite sequences by

T: (a_k)_{k=-\infty}^\infty \mapsto (a_{k+1})_{k=-\infty}^\infty.

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

S: (a_1, a_2, a_3, \ldots) \mapsto (0, a_1, a_2, \ldots)

and on two-sided infinite sequences by

T^{-1}:(a_k)_{k=-\infty}^\infty \mapsto (a_{k-1})_{k=-\infty}^\infty.

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

Abelian groups

In general, as illustrated above, if $F$ is a function on an abelian group $G$ , and $h$ is an element of $G$ , the shift operator $T g$ maps $F$ to^[6]^[7]

F_{g}(h)=F(h+g).

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 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 $ℓ 2 (Z)$ . The shift operator acting on functions of a real variable is a unitary operator on $L 2 (R)$ .

In both cases, the (left) shift operator satisfies the following commutation relation with the Fourier transform:

\mathcal{F} T^t = M^t \mathcal{F},

where $M t$ is the multiplication operator by $exp(i t x)$ . Therefore, the spectrum of $T t$ is the unit circle.

The one-sided shift $S$ acting on $ℓ 2 (N)$ is a proper isometry with range equal to all vectors which vanish in the first coordinate. The operator S is a compression of T⁻¹, in the sense that

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

where $y$ is the vector in $ℓ 2 (Z)$ with $y i$ = $x i$ for $i \geq 0$ and $y i$ = $0$ for $i < 0$ . This observation is at the heart of the construction of many unitary dilations of isometries.

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

Generalisation

Jean Delsarte introduced the notion of generalised shift operator (also called generalised displacement operator); it was further developed by Boris Levitan.^[2]^[8]^[9]

A family of operators ${L x} x \in X$ acting on a space $Φ$ of functions from a set $X$ to $C$ is called a family of generalised shift operators if the following properties hold:

Associativity: let $(R y f)(x)$ = $(L x f)(y)$ . Then $L x R y$ = $R y L x$ .
There exists $e \in X$ such that $L e$ is the identity operator.

In this case, the set $X$ is called a hypergroup.

Notes

↑ Weisstein, Eric W. "Shift Operator". MathWorld.
1 2 Marchenko, V. A. (2006). "The generalized shift, transformation operators, and inverse problems". Mathematical events of the twentieth century. Berlin: Springer. pp. 145–162. MR 2182783. doi:10.1007/3-540-29462-7_8.
↑ Jordan, Charles, (1939/1965). Calculus of Finite Differences, (AMS Chelsea Publishing), ISBN 978-0828400336 .
↑ 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 , online.
↑ p 75 of Georg Scheffers (1891): Sophus Lie, Vorlesungen Ueber Differentialgleichungen Mit Bekannten Infinitesimalen Transformationen, Teubner, Leipzig, 1891. ISBN 978-3743343078 online
1 2 Aczel, J (2006), Lectures on Functional Equations and Their Applications (Dover Books on Mathematics, 2006), Ch. 6, ISBN 978-0486445236 .
↑ "A one-parameter continuous group is equivalent to a group of translations". M Hamermesh, ibid.
↑ Levitan, B.M.; Litvinov, G.L. (2001) [1994], "Generalized displacement operators", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4
↑ Bredikhina, E.A. (2001) [1994], "Almost-periodic function", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4

Bibliography

Jonathan R. Partington, Linear Operators and Linear Systems, An Analytical Approach to Control Theory, (2004) London Mathematical Society Student Texts 60, Cambridge University Press.
Marvin Rosenblum and James Rovnyak, Hardy Classes and Operator Theory, (1985) Oxford University Press.

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