Shift 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 ⁠${\displaystyle t\in \mathbb{R}}$⁠) takes a function f on ⁠${\displaystyle \mathbb{R}}$⁠ to its translation f_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 ⁠${\displaystyle \frac{d}{dx}}$⁠ was introduced by Lagrange,

${\displaystyle T^t=e^{t\frac{d}{dx}},}$

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,

${\displaystyle \exp \left(t\beta (x)\frac{d}{dx}\right)f(x)=\exp \left(t\frac{d}{dh}\right)F(h)=F(h+t)=f\left(h^{-1}(h(x)+t)\right),}$

where the canonical coordinates h (Abel functions) are defined such that

${\displaystyle h^{\prime }(x)\equiv \frac{1}{\beta (x)},f(x)\equiv F(h(x)).}$

For example, it easily follows that ${\displaystyle \beta (x)=x}$ yields scaling,

${\displaystyle \exp \left(tx\frac{d}{dx}\right)f(x)=f(e^{t}x),}$

hence ${\displaystyle \exp \left(i\pi x\frac{d}{dx}\right)f(x)=f(-x)}$ (parity); likewise, ${\displaystyle \beta (x)=x^2}$ yields

${\displaystyle \exp \left(t{x}^2\frac{d}{dx}\right)f(x)=f\left(\frac{x}{1-tx}\right),}$

${\displaystyle \beta (x)=\frac{1}{x}}$ yields

${\displaystyle \exp \left(\frac{t}{x}\frac{d}{dx}\right)f(x)=f\left(\sqrt{x^2+2t}\right),}$

${\displaystyle \beta (x)=e^x}$ yields

${\displaystyle \exp \left(t{e}^x\frac{d}{dx}\right)f(x)=f\left(\ln \left(\frac{1}{e^{-x}-t}\right)\right),}$

etc.

The initial condition of the flow and the group property completely determine the entire Lie flow, providing a solution to the translation functional equation

${\displaystyle f_t(f_{\tau }(x))=f_{t+\tau }(x).}$

Sequences

The left shift operator acts on one-sided infinite sequence of numbers by

${\displaystyle S^{\ast }:(a_1,a_2,a_3,\dots )\mapsto (a_2,a_3,a_4,\dots )}$

and on two-sided infinite sequences by

${\displaystyle T:(a_k{)}_{k=-\mathrm{\infty }}^{\mathrm{\infty }}\mapsto (a_{k+1}{)}_{k=-\mathrm{\infty }}^{\mathrm{\infty }}.}$

The right shift operator acts on one-sided infinite sequence of numbers by

${\displaystyle S:(a_1,a_2,a_3,\dots )\mapsto (0,a_1,a_2,\dots )}$

and on two-sided infinite sequences by

${\displaystyle T^{-1}:(a_k{)}_{k=-\mathrm{\infty }}^{\mathrm{\infty }}\mapsto (a_{k-1}{)}_{k=-\mathrm{\infty }}^{\mathrm{\infty }}.}$

The right and left shift operators acting on two-sided infinite sequences are called bilateral shifts.

Abelian groups

In general, as illustrated above, if F is a function on an abelian group G, and h is an element of G, the shift operator T ^g maps F to

${\displaystyle F_g(h)=F(h+g).}$