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In
mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, 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 (for example, Inner product space#Definition, inner product, Norm (mathematics ...
, the shift operator, also known as the translation operator, is an operator that takes a function to its translation . In
time series analysis In mathematics, a time series is a series of data points indexed (or listed or graphed) in time order. Most commonly, a time series is a sequence taken at successive equally spaced points in time. Thus it is a sequence of discrete-time data. ...
, 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 mapping V \to W between two vector spaces that pr ...
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 investigating the connections between a function and its representation in frequency. The frequency representation is found by using the Fourier transform for functions on unbounded do ...
, 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. Definition 1 Let \mathbb be the set of real numbers and \mathbb be the set of complex numbers. A function f: \mathbb \to \mathbb is ...
s,
derivative In mathematics, the derivative is a fundamental tool that quantifies the sensitivity to change of a function's output with respect to its input. The derivative of a function of a single variable at a chosen input value, when it exists, is t ...
s, and
convolution In mathematics (in particular, functional analysis), convolution is a operation (mathematics), mathematical operation on two function (mathematics), functions f and g that produces a third function f*g, as the integral of the product of the two ...
. 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) H^p are 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 . In real anal ...
s, the theory of
abelian varieties In mathematics, particularly in algebraic geometry, complex analysis and algebraic number theory, an abelian variety is a smooth projective algebraic variety that is also an algebraic group, i.e., has a group law that can be defined by regular f ...
, and the theory of
symbolic dynamics In mathematics, symbolic dynamics is the study of dynamical systems defined on a discrete space consisting of infinite sequences of abstract symbols. The evolution of the dynamical system is defined as a simple shift of the sequence. Because of t ...
, for which the baker's map is an explicit representation. The notion of
triangulated category In mathematics, a triangulated category is a category with the additional structure of a "translation functor" and a class of "exact triangles". Prominent examples are the derived category of an abelian category, as well as the stable homotopy ca ...
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 Operational calculus, also known as operational analysis, is a technique by which problems in Mathematical Analysis, analysis, in particular differential equations, are transformed into algebraic problems, usually the problem of solving a polynomia ...
representation of the linear operator in terms of the plain derivative was introduced by
Lagrange Joseph-Louis Lagrange (born Giuseppe Luigi LagrangiaTaylor expansion In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor ser ...
in ; and whose action on the monomial is evident by the
binomial theorem In elementary algebra, the binomial theorem (or binomial expansion) describes the algebraic expansion of powers of a binomial. According to the theorem, the power expands into a polynomial with terms of the form , where the exponents and a ...
, 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, : \exp\left(t \beta(x) \frac\right) f(x) = \exp\left(t \frac\right) 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, : \exp\left(t x \frac\right) f(x) = f(e^t x) , hence \exp\left(i\pi x \tfrac\right) f(x) = f(-x) (parity); likewise, \beta (x)=x^2 yields : \exp\left(t x^2 \frac\right) f(x) = f \left(\frac\right), \beta (x)= \tfrac yields : \exp\left(\frac \frac\right) 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 call ...
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 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 call ...
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 commu ...
, 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 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. Non-trivial examples include rotations, reflections, and the Fourier operator. Unitary operators generalize unitar ...
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 In operator theory, a multiplication operator is a linear operator defined on some vector space of functions and whose value at a function is given by multiplication by a fixed function . That is, T_f\varphi(x) = f(x) \varphi (x) \quad for all ...
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'' me ...
with
range Range may refer to: Geography * Range (geographic), a chain of hills or mountains; a somewhat linear, complex mountainous or hilly area (cordillera, sierra) ** Mountain range, a group of mountains bordered by lowlands * Range, a term used to i ...
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 and standardize the position of the points or other geometric elements on a manifold such as Euclidean space. The coordinates are ...
. The operator 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 A spectrum (: spectra or spectrums) is a set of related ideas, objects, or properties whose features overlap such that they blend to form a continuum. The word ''spectrum'' was first used scientifically in optics to describe the rainbow of co ...
of is the
unit disk In mathematics, the open unit disk (or disc) around ''P'' (where ''P'' is a given point in the plane), is the set of points whose distance from ''P'' is less than 1: :D_1(P) = \.\, The closed unit disk around ''P'' is the set of points whose d ...
. 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. 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 In mathematics, the associative property is a property of some binary operations that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a Validity (logic), valid rule of replaceme ...
: let (R^y f)(x) = (L^x f)(y). Then L^x R^y = R^y L^x. # There exists in such that is the
identity operator Graph of the identity function on the real numbers In mathematics, an identity function, also called an identity relation, identity map or identity transformation, is a function that always returns the value that was used as its argument, unc ...
. In this case, the set is called a hypergroup.


See also

*
Arithmetic shift In computer programming, an arithmetic shift is a shift operator, sometimes termed a signed shift (though it is not restricted to signed operands). The two basic types are the arithmetic left shift and the arithmetic right shift. For binary ...
*
Logical shift In computer science, a logical shift is a bitwise operation that shifts all the bits of its operand. The two base variants are the logical left shift and the logical right shift. This is further modulated by the number of bit positions a given v ...
* Clock and shift matrices *
Finite difference A finite difference is a mathematical expression of the form . Finite differences (or the associated difference quotients) are often used as approximations of derivatives, such as in numerical differentiation. The difference operator, commonly d ...
* Translation operator (quantum mechanics)


Notes


Bibliography

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