Editing Shift operator (section)

{{short description|Linear mathematical operator which translates a function}}
{{About|shift operators in mathematics|operators in computer programming languages|Bit shift|the shift operator of group schemes|Verschiebung operator}}

In [[mathematics]], and in particular [[functional analysis]], the '''shift operator''', also known as the '''translation operator''', is an [[Operator (mathematics)|operator]] that takes a [[Function (mathematics)|function]] {{math|''x'' ↦ ''f''(''x'')}}
to its '''translation''' {{math|''x'' ↦ ''f''(''x'' + ''a'')}}.<ref>{{MathWorld|id=ShiftOperator|title=Shift Operator}}</ref> In [[time series analysis]], the shift operator is called the ''[[lag operator]]''.

Shift operators are examples of [[linear operator]]s, important for their simplicity and natural occurrence. The shift operator action on [[Function of a real variable|functions of a real variable]] plays an important role in [[harmonic analysis]], for example, it appears in the definitions of [[almost periodic function#Uniform or Bohr or Bochner almost periodic functions|almost periodic functions]], [[positive-definite function]]s, [[derivative]]s, and [[convolution]].<ref name=mar>{{cite book|mr=2182783|last=Marchenko|first=V. A.|author-link=Vladimir Marchenko|chapter=The generalized shift, transformation operators, and inverse problems|title=Mathematical events of the twentieth century|pages=145&ndash;162|publisher=Springer|location=Berlin|year=2006|doi=10.1007/3-540-29462-7_8|isbn=978-3-540-23235-3 }}</ref> Shifts of sequences (functions of an integer variable) appear in diverse areas such as [[Hardy space]]s, the theory of [[abelian variety|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 [[categorification | categorified]] analogue of the shift operator.