In mathematics, and in particular functional analysis, the shift operator, also known as the 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, 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 is an explicit representation. The notion of triangulated category is a categorified analogue of the shift operator.
Definition
Functions of a real variable The shift operator T t (where t ∈ R {\displaystyle t\in \mathbb {R} } ) takes a function f on R {\displaystyle \mathbb {R} } to its translation ft,
T t f ( x ) = f t ( x ) = f ( x + t ) . {\displaystyle T^{t}f(x)=f_{t}(x)=f(x+t)~.}
A practical operational calculus representation of the linear operator T t in terms of the plain derivative d d x {\displaystyle {\tfrac {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 in Heaviside's calculus. The operator thus provides the prototype for Lie's celebrated advective flow for Abelian groups,
exp ( t β ( x ) d d x ) f ( x ) = exp ( t d d h ) F ( h ) = F ( h + t ) = f ( h − 1 ( h ( x ) + t ) ) , {\displaystyle \exp \left(t\beta (x){\frac {d}{dx}}\right)f(x)=\exp \left(t{\frac {d}{dh}}\right)F(h)=F(h+t)=f\left(h^{-1}(h(x)+t)\right),}
where the canonical coordinates h (Abel functions) are defined such that
h ′ ( x ) ≡ 1 β ( x ) , f ( x ) ≡ F ( h ( x ) ) . {\displaystyle h'(x)\equiv {\frac {1}{\beta (x)}}~,\qquad f(x)\equiv F(h(x)).}
For example, it easily follows that β ( x ) = x {\displaystyle \beta (x)=x} yields scaling,
exp ( t x d d x ) f ( x ) = f ( e t x ) , {\displaystyle \exp \left(tx{\frac {d}{dx}}\right)f(x)=f(e^{t}x),}
hence exp ( i π x d d x ) f ( x ) = f ( − x ) {\displaystyle \exp \left(i\pi x{\tfrac {d}{dx}}\right)f(x)=f(-x)} (parity); likewise,
β ( x ) = x 2 {\displaystyle \beta (x)=x^{2}} yields
exp ( t x 2 d d x ) f ( x ) = f ( x 1 − t x ) , {\displaystyle \exp \left(tx^{2}{\frac {d}{dx}}\right)f(x)=f\left({\frac {x}{1-tx}}\right),}
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