Shift operator

In mathematics, and in particular functional analysis, the shift operator or translation operator is an operator that takes a function x ↦ f(x) to its translation x ↦ f(x + a). 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, 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 is an explicit representation.

The shift operator T^t (t ∈ R) takes a function f on R to its translation f_t ,

A practical representation of the linear operator T^t in terms of the plain derivative ^d⁄_dx was introduced by Lagrange,

which may be interpreted operationally through its formal Taylor expansion in t; and whose action on the monomial xⁿ is evident by the binomial theorem, and hence on all series in x, and so all functions f(x) as above. This, then, is a formal encoding of the Taylor expansion.

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