---
tag: 8E0EBA8X07
kind: concept
name: Forward difference operator
msc: [39A70, 39A12]
specializes: [7JSQZ126BP]
sources:
  - url: https://en.wikipedia.org/wiki/Finite_difference
---

## Summary

The forward difference operator $\Delta_h$ sends a function $f$ to the function $x \mapsto f(x + h) - f(x)$.
It is the discrete analogue of differentiation, and difference equations are often written with it.
It equals $E_h - I$, where $E_h$ is the [[shift operator|MBK6AE6734]] and $I$ is the identity.

## Explanations

### L1 human

The forward difference looks one step ahead and records how much the function changes.
For a sequence $u_0, u_1, u_2, \dots$ with step $1$ it gives the sequence of gaps $u_1 - u_0, u_2 - u_1, \dots$.
For the squares $0, 1, 4, 9, 16, \dots$ the gaps are the odd numbers $1, 3, 5, 7, \dots$.
Doing it again gives the constant gaps $2, 2, 2, \dots$, just as the second derivative of $x^2$ is the constant $2$.