---
tag: MBK6AE6734
kind: concept
name: Shift operator
msc: [39A70, 47B39]
sources:
  - url: https://en.wikipedia.org/wiki/Shift_operator
same_as:
  wikidata: Q2278680
  wikipedia: Shift operator
---

## Summary

A shift operator moves the argument of a function or the index of a sequence by a fixed amount.
On functions of a real variable it sends $f$ to $x \mapsto f(x + h)$; on sequences it sends $(u_n)$ to $(u_{n+1})$.
It is the building block of the operator calculus of finite differences, since the forward difference is the shift minus the identity.
In functional analysis the shift on sequence spaces is a basic example of a linear operator.

## Explanations

### L1 human

Shifting a sequence means dropping its first term and moving every other term one place to the left.
The sequence $1, 4, 9, 16, \dots$ becomes $4, 9, 16, 25, \dots$.
For a function, shifting by $h$ moves its graph $h$ units to the left.
A linear recurrence such as $u_{n+2} = u_{n+1} + u_n$ can be written with the shift $E$ as $(E^2 - E - I) u = 0$, a polynomial in $E$ applied to $u$.