Shift operator explained

In mathematics, and in particular functional analysis, the shift operator, also known as the translation operator, is an operator that takes a function to its translation . 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, derivatives, and convolution.^[1] 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. The notion of triangulated category is a categorified analogue of the shift operator.

Definition

Functions of a real variable

The shift operator (where) takes a function on to its translation,

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

A practical operational calculus representation of the linear operator in terms of the plain derivative was introduced by Lagrange,

which may be interpreted operationally through its formal Taylor expansion in ; and whose action on the monomial is evident by the binomial theorem, and hence on all series in, and so all functions as above.^[2] This, then, is a formal encoding of the Taylor expansion in Heaviside's calculus.

The operator thus provides the prototype^[3] for Lie's celebrated advective flow for Abelian groups,

\exp\left(t \beta(x) (d)/(dx) \right) f(x) = \exp\left(t (d)/(dh) \right) F(h) = F(h + t) = f\left(h ^{-1} (h(x) + t)\right), where the canonical coordinates (Abel functions) are defined such that h'(x)\equiv (1)/(\beta(x)) ~, f(x)\equiv F(h(x)).

For example, it easily follows that

\beta (x) = x yields scaling,\exp\left(t x (d)/(dx) \right) f(x) = f(e ^{t} x) , hence \exp\left(i\pi x \tfrac{d}{dx}\right) f(x) = f( - x) (parity); likewise, \beta (x) = x ^{2} yields^[4] \exp\left(t x ^{2} (d)/(dx) \right) f(x) = f \left( (x)/(1- tx) \right), \beta (x) = \tfrac{ 1 }{x} yields \exp\left( (t)/(x) (d)/(dx) \right) f(x) = f \left(\sqrt{x ^{2+2t}} \right) , \beta (x) = e ^{x} yields \exp\left (t e ^{x} (d)/(dx) \right ) f(x) = f\left ( ln \left ( (1)/(e ^{-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^[5]

f _{t(f} _{\tau} (x)) = f _{t+\tau} (x) .

Sequences

See main article: Shift space. 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)} ^{inf ty} ^{k =-inf ty} \mapsto (a _{k+1} ) ^{inf ty.} _{k =-inf ty}

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)} ^{inf ty} ^{k =-inf ty} \mapsto (a _{k-1} ) ^{inf ty.} _{k =-inf ty}

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

Abelian groups

In general, as illustrated above, if is a function on an abelian group, and is an element of, the shift operator maps to^[6]

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 The shift operator acting on functions of a real variable is a unitary operator on

In both cases, the (left) shift operator satisfies the following commutation relation with the Fourier transform:$$\backslash mathcal\; T^t\; =\; M^t\; \backslash mathcal,$$where is the multiplication operator by . Therefore, the spectrum of is the unit circle.

The one-sided shift acting on is a proper isometry with range equal to all vectors which vanish in the first coordinate. The operator is a compression of, in the sense that$$T^y\; =\; Sx\; \backslash text\; x\; \backslash in\; \backslash ell^2(\backslash N),$$where is the vector in with for and for . This observation is at the heart of the construction of many unitary dilations of isometries.

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

Generalization

Jean Delsarte introduced the notion of generalized shift operator (also called generalized displacement operator); it was further developed by Boris Levitan.^[1]

A family of operators acting on a space of functions from a set to 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.}

1. There exists in such that is the identity operator.

In this case, the set is called a hypergroup.

See also

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

Bibliography

• Book: Partington, Jonathan R. . Linear Operators and Linear Systems . Cambridge University Press . March 15, 2004 . 978-0-521-83734-7 . 10.1017/cbo9780511616693.
• Marvin Rosenblum and James Rovnyak, Hardy Classes and Operator Theory, (1985) Oxford University Press.

Notes and References

1. Book: Marchenko, V. A.. 2182783. Vladimir Marchenko. The generalized shift, transformation operators, and inverse problems. Mathematical events of the twentieth century. 145 - 162. Springer. Berlin. 2006. 10.1007/3-540-29462-7_8. 978-3-540-23235-3 .
2. Jordan, Charles, (1939/1965). Calculus of Finite Differences, (AMS Chelsea Publishing), .
3. 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.
4. p 75 of Georg Scheffers (1891): Sophus Lie, Vorlesungen Ueber Differentialgleichungen Mit Bekannten Infinitesimalen Transformationen, Teubner, Leipzig, 1891. online
5. Aczel, J (2006), Lectures on Functional Equations and Their Applications (Dover Books on Mathematics, 2006), Ch. 6, .
6. "A one-parameter continuous group is equivalent to a group of translations". M Hamermesh, ibid.