mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, and in particular functional analysis, the shift operator also known as translation operator is an operator that takes a function to its translation . In time series analysis, the shift operator is called the

lag operator In time series analysis, the lag operator (L) or backshift operator (B) operates on an element of a time series to produce the previous element. For example, given some time series :X= \ then : L X_t = X_ for all t > 1 or similarly in term ...

. Shift operators are examples of

linear operator In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pre ...

s, important for their simplicity and natural occurrence. The shift operator action on functions of a real variable plays an important role in

harmonic analysis Harmonic analysis is a branch of mathematics concerned with the representation of Function (mathematics), functions or signals as the Superposition principle, superposition of basic waves, and the study of and generalization of the notions of Fo ...

, for example, it appears in the definitions of

almost periodic functions In mathematics, an almost periodic function is, loosely speaking, a function of a real number that is periodic to within any desired level of accuracy, given suitably long, well-distributed "almost-periods". The concept was first studied by Haral ...

, positive-definite functions, derivatives, and convolution. 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 In dynamical systems theory, the baker's map is a chaotic map from the unit square into itself. It is named after a kneading operation that bakers apply to dough: the dough is cut in half, and the two halves are stacked on one another, and comp ...

is an explicit representation.

Definition

Functions of a real variable

The shift operator (where ) takes a function on R 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 Joseph-Louis Lagrange (born Giuseppe Luigi LagrangiaTaylor 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. This, then, is a formal encoding of the Taylor expansion in Heaviside's calculus. The operator thus provides the prototype for Lie's celebrated advective flow for Abelian groups, :

e^ f(x) = e^ F(h) = F(h+t) = f\left(h^(h(x)+t)\right),

where the canonical coordinates ( Abel functions) are defined such that :

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

For example, it easily follows that

\beta (x)=x

yields scaling, :

e^ f(x) =   f(e^t x) ,

hence

e^ f(x) = f(-x)

(parity); likewise,

\beta (x)=x^2

yields :

e^ f(x) = f \left(\frac\right),

\beta (x)=1/x

yields :

e^ f(x) = f \left(\sqrt \right) ,

\beta (x)=e^x

yields :

\exp\left (t e^x \frac d  \right ) f(x) = f\left (\ln \left (\frac \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 equationAczel, J (2006), ''Lectures on Functional Equations and Their Applications'' (Dover Books on Mathematics, 2006), Ch. 6, . :

f_t(f_\tau (x))=f_ (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)_^\infty \mapsto (a_)_^\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^:(a_k)_^\infty \mapsto (a_)_^\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 is a function on an abelian group , and is an element of , the shift operator maps to :

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 In functional analysis and operator theory, a bounded linear operator is a linear transformation L : X \to Y between topological vector spaces (TVSs) X and Y that maps bounded subsets of X to bounded subsets of Y. If X and Y are normed vect ...

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:

\mathcal T^t = M^t \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 ''S'' is a compression of , in the sense that

T^y = Sx \text x \in \ell^2(\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 ''S'' is the unit disk. The shift ''S'' is one example of a Fredholm operator; it has Fredholm index −1.

Generalization

Jean Delsarte Jean Frédéric Auguste Delsarte (19 October 1903, Fourmies – 28 November 1968, Nancy) was a French mathematician known for his work in mathematical analysis, in particular, for introducing mean-periodic functions and generalised shift ...

introduced the notion of generalized shift operator (also called generalized displacement operator); it was further developed by Boris Levitan. 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: # Associativity: let . Then . # There exists in such that is the identity operator. In this case, the set is called a hypergroup.

Notes

Bibliography

* * Marvin Rosenblum and James Rovnyak, ''Hardy Classes and Operator Theory'', (1985) Oxford University Press. {{DEFAULTSORT:Shift Operator Unitary operators

Definition

Functions of a real variable

Sequences

Abelian groups

Properties of the shift operator

Action on Hilbert spaces

Generalization

See also

Notes

Bibliography