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 Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. inner product, norm, topology, etc.) and the linear functions defi ...

, the shift operator also known as translation operator is an operator that takes a function to its translation . In

time series analysis In 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 m ...

, the shift operator is called the lag operator. 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 Map (mathematics), mapping V \to W between two vect ...

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 functions or signals as the superposition of basic waves, and the study of and generalization of the notions of Fourier series and Fourier transforms (i.e. an ex ...

, for example, it appears in the definitions of almost periodic functions,

positive-definite function In mathematics, a positive-definite function is, depending on the context, either of two types of function. Most common usage A ''positive-definite function'' of a real variable ''x'' is a complex-valued function f: \mathbb \to \mathbb such ...

derivative In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. ...

s, and

convolution In mathematics (in particular, functional analysis), convolution is a mathematical operation on two functions ( and ) that produces a third function (f*g) that expresses how the shape of one is modified by the other. The term ''convolution'' ...

. Shifts of sequences (functions of an integer variable) appear in diverse areas such as

Hardy space In complex analysis, the Hardy spaces (or Hardy classes) ''Hp'' are certain spaces of holomorphic functions on the unit disk or upper half plane. They were introduced by Frigyes Riesz , who named them after G. H. Hardy, because of the paper . I ...

s, the theory of

abelian varieties In mathematics, particularly in algebraic geometry, complex analysis and algebraic number theory, an abelian variety is a Algebraic variety#Projective variety, projective algebraic variety that is also an algebraic group, i.e., has a group law th ...

, 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 compr ...

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 In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is called t ...

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

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 In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is comm ...

, 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 vec ...

with norm one.

Action on Hilbert spaces

The shift operator acting on two-sided sequences is a

unitary operator In functional analysis, a unitary operator is a surjective bounded operator on a Hilbert space that preserves the inner product. Unitary operators are usually taken as operating ''on'' a Hilbert space, but the same notion serves to define the co ...

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 In mathematics, an isometry (or congruence, or congruent transformation) is a distance-preserving transformation between metric spaces, usually assumed to be bijective. The word isometry is derived from the Ancient Greek: ἴσος ''isos'' ...

with range equal to all vectors which vanish in the first

coordinate In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine the position of the points or other geometric elements on a manifold such as Euclidean space. The order of the coordinates is sign ...

. 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 o ...

introduced the notion of generalized shift operator (also called generalized displacement operator); it was further developed by

Boris Levitan Boris Levitan (7 June 1914 – 4 April 2004) was a mathematician known in particular for his work on almost periodic functions, and Sturm–Liouville operators, especially, on inverse scattering. Life Boris Levitan was born in Berdyans ...

. 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